When come on everybody to the lecture on Estimation and Inference,
we had our first licture today with first review of prombility theory,
I will continue with this today,
may I ask you if there are any questions than please
raise your hand and type in the question in the chat
I don't see any questions, so so let me continue,
what we did, yes today,
was basically that we defined, what a probility,
dentity planction and cumulate that distribution function, is,
the this functions descuscribe,
actually completely the behavior of random variable or the properties as should say,
of a random variable,
but very often we are not really interested in the conclete set of characteristics of such a random variable,
but only in a view aspects of the distribution.
In concepts as for instance the expected value or the Variance,
or the standard diviation of the random variable as measure of its a variability,
or the correlation, one variable has with different random variable,
correlation or converiance.
So it this second subsection of chepter one,
I will focus on such concepts
Now you certain you are know,
what the expected value of a rando variable is,
but I would still give you formal definition obbit, the intuition is,
of course, that and expected value of a rando variable,
is awayted average of the possible outcomes.
So of the values,
switch the variable ex takes with certain proabilities,
of these values when takes awaited average,
and the waits are the proabilities,
which are signed to the random variable in the case of
a discreetrain and variable,
or the Density,
which is societed with itch possible outcome of the random very
The definition of the expected value here,
distinguishes between the two cases, which we discuss yesterday,
discreed random verible or continuous random verible,
if we first look at the case of a discreed random verible ex,
with as I have sitted, out yesterday,
a cape possible outcomes, ex one up to excape,
then the expectation of the random verible,
that is to say the expected value of the random verible,
is defined as the waited some of these outcomes.
So you see the outcomes here,
ex one and ex two,
extreme up to exca
Each of those access being multiplied by well F of ex one
the value of the proabability identity function at point ex one
or ex two ex three or ex cay and your remember the
F of ex one is just the proabability for event ex one or curring as one outcome of the random experiment which the random very build this cribs.
So we could also right this down as ex one times
p1 plus ex two times pe to as plus plus ex
cay types pe you the sociated probility of outcome that
ex king
Kind of course, also reacted in this way by using the
signal science to invidicate a some consisting here of pay terms,
extray and fhemd might plate by in F of extray.
This way of writing it is helpful, because then it is
most suggested, to see the analygic to the continuous case.
So when ex is a continuous random variable,
then the expected value of ex is just defined as the ideodom,
over ex times if of ex the ex
And you know probably that the intergram,
which looks like a stylized capital s, the rives,
it's form basically from the idea that integration,
integration is nothing else than summation,
that's what's the s here,
this would be a greek as the letter sigma and this here,
would be a stylized roman as for some,
the difference between the two ways of someming terms being that the sigma is used,
when we have countably many terms to some,
so I've a
finitely many or infinitely many but
still countable many
And the integral is huged when the number of infinant elements is not countable anymore.
And then sure if you sinnery call to definition of countability from Mathematics,
it's not big issue if you do not recall this award explained,
now but basically what you dust have to remember is that
in the case of discreed very abless used the Capital Sigma to some and in the case of a continuous very used.
The integral both of which will essentially for full the same faction,
of something the product of the possible outcomes here ex,
waited by F of X. And this case in the discreed case,
the access are indext and so are the social atted probilities,
F of XG, so that here the indext jay is used,
in the case of continuous veryables,
we cannot index this anymore or the standard motation at least is not to index is it,
but rather to discribe,
but the term,
the ext that ex moves smoothly over the whole intervall,
of a which the internal is defined.
And it is defined for the expected value always,
as be integral for negative infinity to plus infinity,
so over all real nevers.
And you know that, that are
all incountably many real members
Okay, this expected Value is formally in terms of mathematics, linear operator.
And again, probably most of you are not family with
operator theory. So I would it's this done that,
and I would want introduce you to operator theory,
this is a mathe medical concept,
which is useful and advanced statistics and economentrics,
but not get very useful or hipful at the level at which at iteach this facture.
So dust moded, it's call the linear operator and linear operator,
linear operators have the property, that the usual ruled of linear functions,
apply to the operator two. So the expectation,
the expective value of some constant a times randomary,
with express some constant be times random variable,
why the expectation of this some here this the
same thing as a types the expectation of ex plus
be times the expectation of why
Offen we will not use the notation, expected value of ex,
as e of ex, but right that apprevious this little bit
bit writing, new ex. So using the greekletter new
for the expectation and then just indexing by the random variable to which this refers,
this case ex, would indicate to us,
that this is the expected value of random variable ex.
This expected value is sometimes also call the population mean,
meaning that, if we really had information about
the whole population of possible events, and we would know
that mean in this population is exactly the expected,
value, where as what we actually have
interns of information is to pictly as sample,
drawn from the population and the everythmatic mean of this particular sample,
is then called the sample, mean.
The sample mean is of course, not necessarily identical to
the population mean, quite to the country, usually the sample mean
devience from the population mean and that is to say
from the expected value of the random carry
Let's look it our, point, possing experiment to CEO,
the expected value would be computed for this experiment, we had this example.
Yes, that they and we analysted it and found out that the probibility
density function of the coin tossing experiment has the following step vice form.
So the probility F of X is point 1 to 5,
if exass equal to zero,
so if the number of heads toast with 3 toastes of the coin is zero,
we don't to those any head, it is point 375,
a for exassing pretty one, point 3 its,
375, for expecutive do.
And then another point 1 two fight,
if exass equal to 3, so 12,
point 5, percenters, the probibibility,
of tossing 3 three heads in three toastes of a fair.
Toll. Any other value is impossible,
so in this is as a cide of roundability of zero
for any other ex of this base of real numbers
There for a easy to compute the expected Value of X,
because this is in discreed random variable.
So all we have to do is back,
we might deply the outcomes zero one to three with the respective probabilities.
And then at the whole thing up.
So we would have have outcome zero for X,
I has proability of 12,5% procent.
So this is one you might deplayed.
Obviously this is irrelevant, because zero time something is always.
We have one for the outcome accessible to one with the proability of point 3,75. So one times point 3,75,
so of course, point 3,75, plus them two,
the outcome that we would tos two quite, two,
or hands in three tosses of the point, two times it's proability,
point 3,75, plus then, the last outcome in three, times
the proability of tossing three, heads in three tosses, are which is 12,5 Prozent. So we have three times zero point one,
two 5. And if we add this on up,
then we are rive at one point 5. And this is of course,
I'm moved very intuitive,
if a head ass you before hand, well,
if you toss the coin, three times and you count,
number of heads, which you will have,
what you are think will be sort of the mean value of hand switch,
are being tost in great number of repetitions,
of this experiment, we would would probably have said,
well,
it is possible to tos zero heads and one two and
three and obviously the recycle,
which is most likely,
would be one point
Okay. You also, I zoom, have a good
at least intuitive other standing of the concepts of Variance and entered dividation,
which are both method measures of variability. And as you know,
the standard dividation is simply this queer root of a Variance, come times,
at this more use for to express the very opility in terms of Variance is,
sometimes it's more use for to express this in terms of sendered diviations,
but they convey exactly the same information.
So basically, it's sent diviations just the flipside of
the Variance
Now, one the next slide, you'll see a graph,
which I capit prom, mold ridge,
which shows the probility intensity functions of two random variables ex and why,
which had to say mean,
but they have different very the ability.
And here this, it see the mean of this variable
of these random variables, is just given by this new here.
So this put be I that, new ex or,
new why, and any case, a new is both mew ex and new why.
So random variables ex and why have the same mean,
but it is you see the very ability is different, the random
variable eve of ex is mean,
that is you see the very ability is different, that random
variable eve of ex is centered more strongly around new
than very able why, this centered around use.
So this random very wise brands out substantially more,
then random variable ex does.
And this is expressed by the concept of a greater variance of random
variable why, or greater standard deviation of random very why,
which is the
House this fromly defined. Will as I already said,
for any randomarible, exwed you not the expected value by
now ex, which is the expected Value of ex.
And then the Variance is denoted by V of ex.
So we of exists the general simple are used from
a varietes of a randomarible,
we of ex is defined to we the expectation of ex,
there are an a variable.
Myers it's expectation new ex squared
Think the expectation or a squeed term and the term,
which we consider, is the deviation of the random variable, exile.
Outcomes of the randomaryable ex, from the expected value of ex,
we take the square of this deviation,
because it wouldn't make sense to add up, the simple deviations,
since we would have positive deviations and negative deviations and the post
it negative deviations would prescisely cancel each other in the symetric distributions.
So then come to the conclusion to the rock
and clusion that dispersion of the randomary is zerope,
which is of course not the case.
This is why we take just positive values,
and there would be many different ways to provide for just positive values,
one way is to look at the square of the deviation.
Principle what could also look at the absolutely value,
but this slash comment or we could look it.
And you other fourth old order tonly nomial, or cubic,
non- of cubic terms, but terms race to the power of four,
68 and so fourth, which would always you've them,
posted element
see, but this is also ancoman,
what is in the common measure is
the square of the deviation of ex from its expected fair
This Variance B of X is all offen,
denoted as is simple, sing my square with index ex,
so always with the index referring to the random variable, who's variance, we talk about.
And the square is then indicating that retect the expectation of a square,
square, term,
on that is is the expectation of a square term,
we as if we bert to look at this termer diviation,
than we would have to take the square root of this term and then,
we would just right,
sing makes,
for the standard diviation,
where seeing my square and exist indeplitation,
for the variance
Now, there's an important to property for variances,
which you have certainly autocoverty your and a grediot statistics.
Thenkly that the variance, which is,
as as I have sent here,
formerly defined as the expectation of ex-minus new ex-squeent.
So calification, refers to the squeent turn.
This variance, we can also and so what easier,
express as the expectation of
ex squeent
Meines, die Expectation of X squired.
So minus the square of the hold expectation of hex,
that is to say, minus new ex squired.
So these two things this expression here and this expression here
are exactly identical for all random variables,
and it's easy to show this, you see
this here the proof is trying here,
when you start out would the expectation of ex-minus musex
and prenticies squired than you can up sorry
Than you can multiply out and this square turn here,
you would get exquered, when this two,
ex, new ex plus, mu ex squared.
This is nothing, no, matic, just multiplying out,
this turn in print to seas here.
And then by the property that the expectation is lineare opparator,
we could also right this as e of ex squared.
Meines two times, e of ex,
X. The mew engstes revemember is a constant.
This is just in real number. This is just
the expectation of exemplication of excessibly a real number.
So it's real number like the two.
And we know from the providy of the linerar opparator,
that such constants can we pulled out of the expectation terms, term.
So this is why we have the two and the mew
X or any outside of the explication operator.
And the expectation operator is then applied to the randberry the X,
of which we then know that this hold things nothing else,
but mew X.
So essentially, he we have minus
two times new X screened
And then we add another term new ex squad,
which comes from this complement here.
So we have negative to you exquered, plus new exquered,
that is of course equal two negative new ex squad.
Since minute exquered is the senthing as your ex square,
the whole expected values great,
we can alsorited
this problem
That the variance of arrandobariable is, I've wirklich a all those always positive.
Well, when you look at this term, here is clear why,
because ex-minus mui ex-scrates is typically something now negative.
Henvizero, can be positive. But if ex is a random variable,
but you would expect that their our certain events,
we are ex is different from uX. And if ex is different from new X,
then we would have a little nation here,
which raise to the power of two,
gives or ice to some positive value.
And there forces, we just to positive or zero values for here,
the expectation operator of this term here,
would yeard a positive, expectation of the scratterm,
therefore the variance should actually be positive be positive,
when every we have a real random variable,
you which sometimes deliates with non-sero probability from
Die Converse-Setting would be that ex is always equal to unique.
When ex is always equal to new ex than obviously
ex ex modus muse ex good be equal to Zero,
as queer up Zero as still Zero,
the expectation of Zero good, also be Zero,
this case we would have a variance, which is Zero.
So I territs for a randomary, which is Zero. But actually,
there would be nothing, which is very,
which is random of that is friend a very build any or,
because we would know for sure that the randomary,
the ex text,
the value new export each realization,
which it may can takes or that would be boring,
this would be what we call
it degenerate randomary
There is a case which is little more intry cat,
because there also a random variables, which have a variants off.
Zero that would give you example of that in the minute.
But first, I would like to know that we
can think of it constant as a generate random variable,
as I'm just sent.
So that would be a random variable,
which has not any a variation at all.
And therefore the variants of such a constant is
Which might lead as to complude,
that we have we of ex equal to Zero,
if and only of there's a constant see,
such that the problembility of ex being equal to see,
is equal two one.
When this looks like by a saying,
the probability of ex being equal to see,
is equal to one, this equal to 100 prozent.
This looks like, we discribe here just a concept.
Rentive the randombarriable takes, the same vel see and each each realization,
then this would mean that divirenemberry, but see is concept.
And this case, we can of course, safe,
we say,
at the probility of ex equal
to see isical too
How ever that are is a said,
more interecated cases of random variables,
which do not have the property that access equal to see for all realizations,
because it's possible that we have a random variable,
which has realist, realizations different from see on,
would is called a set of measure zero.
The set of measure zero, means that we have some
subset and the set on which the randomberry is defined.
And the probably biblity that some event or curse,
which is drawn from this subset, is simply zero.
Never the less, this subset is not empty.
Too give you just one example, if you think of
on integres for instance, then you know than on integres
or just countably many. I mean, there are infernitely
many, as you know, but you can come.
And you can make a list of all the integress. Where is the real numbers are more than countably? Many,
and this leads to the result that more terms of measure the ory,
the set of integre in the set of real lambers,
is of measure zero, basically the probablity that when you are draw,
when you hit straight line,
which you think of as the the picking the real numbers,
when you hit is string line,
the proability that you hit and integre, is basically service.
This is also mathematically, it is just zero. So,
continuous random verabled, which have,
I that fine ed number of results of the results of the random variable,
which deve it from see,
all which have just countably many such events,
we are the result biblioth from see, would have some very ability,
because on ever this some circle of stances,
the result of the random very,
will would not be ecould to seems, but something eads,
but the probablemability of this results of curring esactions zero.
And there for, all we can sayou say,
that probility of exploty equal to seeat,
is actions
to firoo
Okay, that's for it for the very it.
Let's talk about the standard deviation of a very bit.
As a said origin, this standard deviation, which we do not sigma X.
So fust it's que, is missing price,
which is be all you thinking between a that deviation and
the simple we use for the variants of a random variable.
The standard deviation sigma, extraverandom very bit extrems,
is this queer root of experience.
So sigma extress defined is to squeer wood off,
be of ex
Ja, with these settings,
we can turn back to the coin tossing period,
and can now a compute the variance of the coin of the random very,
the ex-which costs the number of heads,
which have been toast with the fair coin
Early in dislecture, we have computed, that the expected
value of excess equal to new ex and that is one.
That this is one point five, this in new ex. So,
new excess one point five, this is deequite,
the expect Value of ex and we make use of this now,
here, by setting up this table.
In the first line of this table,
we just right down the 4 possible outcomes,
for the rand verable ex which counts the numbers of heads,
that have been toast.
So they can be. Zero hits, one head,
two hits on three head. These are the three,
the expect use me, the 4 possible events, which can.
Okay. These for possibly events as you know have probablities,
which we have computed,
not the numbers by now point one to 5 5,
point 3-7 5,
point 3-7 5 and point one to 5
are the PJs of the F of exchase
So we can than compute the difference between extra and the expected value office change.
Nämlich journey the expected Valley of ex. Name the new
ex, which given you hear is one point live.
Then we see that, this difference exchange. New exs
is negative one point fire. Here negative zero 5,
hear is zero point type here and one point 5,
When we squeen this difference, well,
that we just hip square the terms in this line.
Here so we get to two point to 5,
5, two 5. So this here is negative one have.
And this being sweared is positive one 4,
so point to 5. Here we have zero point 5.
So this, also one 5,
and then we have one pon 5 and two point two
And now, we way these terms, the squeed terms with the problem.
But these so we compute, PJ times extreme minus,
buy ex squeed or F of Extray,
as we have label this in the europinionative Edition of the Variance,
could also right here F of Extray times extreme minus,
we're expert this.
The same thing and save the little space by dust lot of them PJ,
well, just computed, zero punt one to 5,
times, two one to 5 is zero point,
two 8,1,
Ed, hier, same thing, happened,
we see the probility up here and the product of the probility with the squeer to diviation from expected,
value, now here, this is then terrible 9,
say think here, and here we can again,
we have this zero point to 8, what,
to 5. Doll we have to do now is to some these for terms here,
and if we do it, you can't verify,
that we do get insectly zero point, 7,
5 or 3 over 4.
So, three forth is the Variance of the random very,
which constant numbers of heads in the contossing experiment
Singen wir. Ex-Seare is than the standard deviation. So
pake the square root of three over 4, well,
the square root of one over 4 is one have and the cremazed square root of.
Free, there for the standard devition is why have a square
root of 3. We can then pake and that this variant computed this way is this same,
which would be result if we used other formula or computing avariants.
This was the expectation of exquered minders the expectational ex,
being squid,
so minders mew ex squad, when we look at,
this that we take the expectation of exquered,
which is zero point 3,
7,5,
Das Zero-Point 375. Times 4, plus Zero-Point one to
5 times. Nein, nein. So,
wir sehen hier always gesquee at resets, right,
one in squee at one,
two being screen to far,
three being squeen to 9. And then we have to substract minus one point 5,
squee at this resets than in Zero-Point
75 or three quarters
No, another important property,
this time referring to random variables and their varities,
if ex is a random variable and AMBR constance,
such that we can,
did live new randomberical set,
that being equal to a plus bx,
then the variance of set is eigentlich bsqueed,
times the variance of ex
So, wie Adasn't Play in your role,
because we subtract the mean, anyway before,
we raised the term to the power.
Too. So, with subtract to the mean this Agos
away, but the beat is not go to way.
And therefore does the variance of set,
depend on be in is squared sense.
So, be squared times the variance of ex
The proof of this is also easy, I show to you,
here, vehe of set is equal two the expectation of
sectiminest e of set that's definition of the variance.
And then that, we know, is a plus a GX.
So we have the expectation of a plus bX meines,
the expectation of the expectation of a plus be X.
And this soul
thing squad
When this is equal two in the exploitation of a plus BX,
which comes progreed, minus A, minus B-Apps,
we tapps the expectation of ex at this thing squid.
And this can be simply fied to year the expectation of BX,
minus, be that's the exectation of ex squid.
And this is the same thing,
of course, is peace quint e-times extremers,
e-of ex squid and this the same thing as be
squired times to their ends of ex,
because this year is the definition of
a verits
So we see that variant seas an der linear transformations increase
by the swear of the factor by which it is being multiplayer.
And the shift in the content term, there's play a role,
the a is not approtent, but the be is quite
deportant because it because it increases the spread of the darns
of ex
Now, for the standard deviation, of course,
sigmans sept is equal toop, be in absolute value,
times see makes. Why in absolute value, now,
because esternard deviation is it a square root of something pastive,
a square root is always defined is me in positive,
typically actually, there would be two solutions studity, respective.
Equation, one positive, one negative solution nostly,
but we are interested only in the positive solution,
in test being defined that as square root is always posted solution to in equision.
So we would have here sigma ell this equal to be
in absolute value types sigma x. Why in absolute value,
now, quite tilly,
we have not made the eascension that be is cost.
We can itly or to be negative,
there are many examples were be would be negative
This doesn't play a role for the formular of the varies of set,
because the be is raise to the power of 2 here sound.
He's queer this always postting. But when we down the
square root, we cannot just stright be here,
because in principle be could be negative.
And then were would have a negative square root or negative
persteller deviation, which is simply not, at missible,
those must and true that be is actually taken just
bys absolutely value and this is why.
We used they absolut
value here
Bei uns sind die Feinden der Co-Variance for You. Für die Co-Variance,
die Concept is, dass wir die too measure,
die too gree of similarity between tourende variables ex and why.
And that these two variables have the expected values new ex and new why,
then the co-Variance of ex and why is defined as
the expectation of ex my use view ex times,
the expectation of why
mond new why
This co-Variance measures the strength of association,
between X and Y,
Velocus visuals, some sense to degree of co-movement,
between X and Y. More recizely it just measures,
the degree of linear
association between X and
Obviously, it is the case that the Variance of X
is just the co-Variance of X with itself.
I think you use this from Unit here, there we have the co-Variance of
X-App Y. And this is equal to the expectation of X minus,
new extent the excitation of white minus, new why.
So if you in certain, ex for my here,
and there, then you would get the same term,
which is just might deplayed by itself so,
there will take this, we are of this term all,
why way the expectation operator,
who take the square of this term,
and therefore,
we would arrive just that
the darians
So, The Variance is a special paste of the convertance.
Now, there are are again, somethings to do it.
The co-Variance can be ritten, a quite inaliguesly to the Variance,
by branding it first as a product. Namly,
e-of ex-minus me, ex-temped why manuss, new way,
and the multiplied out to yeard, tja,
what is a way, to year,
e-of ats-wight times, new ex-wy, myers, sorry,
not times, miners new ex-wy, miners new way,
ex plus new ex,
new by way
So, this simple out multiplication of this. Print to
sees here, there we can pull the expectations operate inside
these print sees here. And might lean, so,
you would have the expectation of ex times. Why here,
my was be ex times, the expectation of why, mys
new wide times, the expectation of ex,
does the product of the two expected values,
new ex and new why.
Now alwaysly eat of why here, is even
two new why. So we would have new ex times
new why
And he we have you why times, new ex.
So accentually, we would have two new ex times to why terms with narrative.
Sign. But he is also a term, which has
positive sign. So it's there a negative two times,
new experience here. That's we are experience we why,
we eventually arrive at narrative new ex times new one.
And this is of course, nothing else,
but he of ex,
why times earph extimes,
your fly
Nur die Ernährung ist,
die Energie to the representation we head for the variants,
where we said, that the variants of ex,
was equal to e of ex square at not party here,
case,
this case,
we have here again,
just at as quite a bit
more general level
Ob so, also please that we of express why,
is eagle two, the expectation of express why squeared,
mine's UX plus plus New Wine Squid.
If we a multiplant is here, out,
there we would get that that we have,
but variants of new express plus way,
being equal two im E-F-Express plus ways que,
mine us new express,
new Wine Squid,
do to this probity up
So, Multiplying aus Expert One in the Square Wise.
Squared, years, exqueered as 2XY, that's wise queered.
And with this turn here, we can than compare the terms which
recycling we you Multiply out new explusions we do are here.
So native new exqueered here, minus two terms new exqueer why,
here, minus new wise squared. Hier is exactly,
what we get way multiply out this term here. Und dann,
obviously, this is a simply the variance of explaced two times the co-Variance of ex and wife,
that's the variance of line, because the expectation of exqueered here,
minus the expectation of ex squared,
so minus muse exqureent,
is pretacly equal to devariance of ex
And the same host for the variants of way. And then there others cross terms,
two times the expectation of ex and way, test expectational,
ex times, way, so tast a proud,
turn, minus 2 times, mu ex mu
way, is just definition of the coverance. So,
a please remember that,
this is really something which you will which need threat often,
the variance of two random variables can me mighty played out,
but what we get then is also the co-variance between ex and way,
in addition to the turture turt what you would expect,
variance of ex plus variance
of my
So it is not the crease, that the variance of express wise,
the same thing is the variance of express the variance of wise.
But but be wrong, we have to at 2 times
beconverance of explain. Now we come two in inequality,
which is called the cushi schwarz, inequality, and which perhaps,
you have not it made in your inner radio.
Studies or webs you have,
but in this case are would expect that props you don't.
Now the proof yet, that you have it seen it.
So since the cushi schwarz, nequality is quite,
studix significant, and economentrics and facisms, as used run afen,
I would like to introduce you to this make only,
if you have seen it, it.
And then auch so provide proof to,
because you shorts,
So, let's ext and why be any random barribles,
then it's holds for anypair of random random barribles,
this about coachy slot says,
then it holds that the co-Variance of ex and duy squiet,
is less then are equal to the proud
of Variances of ex
and why
In a proof of this crish schwarz inequality is not trying.
A provide you here with the simplest proof that I'm love
that are quite a few it. I would say,
it can't three different proofs available for the crish is schwarz inequality,
but this one here is the simplest approve and you'll see it is still
not yet completely simple and we first all to have the
idea on how to proof this.
Because people who would use this proof start out in a
runner other main with the cushi
schwarz quality by saying that we compute the varies of nember
expand wides of nember express wine.
For any lander which this taken from they real enembers.
So take any lember from the real enembers and then computed
the varial enembers and then computed the variance of lamber express
like we are the lamber is just any
real enember
No, from the roots, that we have just discussed.
We know that the variance of lamber ex-pression my is equal
to the expectation of lamber express wise, create miners.
The squared term of londa eof express eof why.
So, this term kino, squared. Well,
we can of course multiply this out,
if we square lamber the express wife,
what we we get, we get londa squared,
exquared plus plus lambler ex times.
Why? Plus wise quered
And of the term, the middle term, lamnder ex times
or why, we have to take the expect to value.
So we would get here lander e of ex, my plus e of wise right.
Time plus even to value of, my squid,
if our this, we would have to subtract,
the lamder squid new exquired and mindless two lamb them you
XBY,
mindless new way squid
Ja, and this can be streckend out, a little bit.
So that we're ripe at londa squared,
we of explus 2 lander, couverriance of ex and Y,
plus variance of, sorry, burns of lying,
and this thing must be greater or eple than Zero why,
because we started out with the barrence,
and we know everrence is always non negative
The two cases here, the various of lander express why
can greater than Zero or can deeple to Zero.
So let as consider first the case of greater them Zero.
What a claim here is that the Variance of lander express
why is actual greater than Zero, unless,
lander express why is a constant with probably one actually which
are made it here, is a constant with problem
D1. So unless it is a generate random variable
It's gotting to hard your said. No,
let us first considered this non-degenerate case with the variance
of lamber explus why is greater than Zero.
Lot ecould to Zero
If the Svarians is greater than Zero, then it is clear that
there is no lander from the real numbers, which selfs
this equivition here. Land us square view of ex,
now, varians of ex plus 2 länder,
co-Varians of ex and Y, plus Varians of Y isequal to Zero.
And to we know at this think you is greater than Zero,
than the parently, there is no lander for which this
equivition here has the so hution in the space of real numbers.
So there is no solution in this lander,
such that this equition here is selbst falsch.
How ever this year is a simple quite ratty equation. And you know,
each quite ratic equasion has two solutions,
but it maybe that the two solutions are in the complex numbers,
of lea and not in the real numbers.
So it just formally computed the solution of this quite ratic equasion in lander,
are you will find that the two solutions to the quite ratic equasion
lander one and lambers two, equal two,
the negative of co-Varians, of ex and way,
over V of X,
class or minus the square root of this
term here
Will, and what do we know now,
when we know that that are no real valued,
so lution to this equasion here,
then apparently this term under the square root must inegot.
And if it very zero or evidual positive,
that we would have real when it's solution.
What know, this term here must be negative in the case,
in which this equasion here doesn't have as solution for lander,
take from the real numbers. So,
under the condition that we of ex is greater than zero,
this thing doesn't have a solution and this would danten
ply, that the term under this square root sign here
is negative
So, for both, earlys of lander, busbly the case,
that co-burns squared, hope on, but cool, their eyes divided by Variance, squared, is smaller then the racial of the two Variances. And when,
and view of one, at the Veh of Y and Veh of X. Okay,
so then we are always already essentially,
we have we want to do to because this is to cutty shots,
inequality, we know this thing here squared,
is less than we of a why over Tee of X,
to from this it follows, ineuitly,
that the co-veriance of X and wise squared,
is less than the Variance of X, that's dodarents of why,
which is, what we cushy shih shots, equality,
still, could shots inequality status.
When, for wearing squared, is less than our equal two,
we of X, we of my,
if not shown for one, non-de-generate, trend and variable,
it is even astrict inequality yearly year, the converian squared,
is less then view of ex-times, veh of light for all mondy generate
rand variables
It's remains to look at the degenerate case,
where lamber ex plus, why is ekulture constant?
Find with some see. This would be the generate case,
where the Variance is constant is zero of their random very lamber ex plus why.
Actually, it's would be a constant with provibility one,
so with 100%. There's still this possibility of fine
atly many or countably many events in set of overly countably many events,
being different from see,
this can be covered exactly all the same line.
See with some edition notation, which adon't give to you,
it is completely sufficient caractly if you,
just considered this case for this simplest
case of a variance of Zero,
we are lam that explus why is
just ekulture constant see
So, if this is true,
that lamb their express why is a constant.
See, then we can take expectations from. Benno,
we can do this, but taking expectations,
we are right at lamb that times the expectation of express
the expectation of why is equal for constant.
See. They are for it holes that lamb that times ex minus muiu
ex, this why minus muiu way is equal to zero
. Or put differently that why mind us new way,
is equal to the negative of lamb the times ex minus mu
ex. How do we know this that this equations here
is zero, where very easy,
we just have to subtract this equation here
From this equasion there. Then on the writing side,
see, my not see, is simply ego to zero.
This is this there are here. And on the left
tend side, we get lander times, ex minus muics,
plus why, by not's mu way.
So, just subtracting this equasion here from this equasion there,
we would get this equasion here.
And we can reright this as as why,
but as new way, is equa two,
legital climbler,
and ex,
minus muics
Before we can easily compute the car variance, now,
the car variance of ex and wise,
equal to the cor variance of ex my thu ex,
I'ms why, and is new like.
And since we know, what like, minus new Y is,
we can a right this as cor variance of ex minus new ex,
and negative lamb the of ex minus new ex,
because negative lamb there, times ex minus new ex,
is speakle
two why minus, new why. And this means that this
is the same thing as the expectation of mague to flamser,
times ex minus new ex squated
And we've then know that we can pull this negative lamter,
out of the expectation, it's operator,
that we have here,
if we have pull it out the expectation of exmails muix,
sweat, which is just the variants.
So we get there said, this is minus lamber times
the variants, of ex. So from this equation,
we see that the col Varians ex of,
to be between ext and wides,
quin it
Is equal to lam us, queer times the variance of ex-forced.
Fust swearing in this equation here, co-variance of except Y,
beings queered, let's queer a right inside of this equasion here,
and the negative tech, negative is positive.
So we remain with landers queered vers of exits,
we at, which is is turn there.
More over, only know, is that, we of my,
it a variance of my, is equal two the
expectation of course of my mind as new Y,
that wait, this is the definition of the variance of my,
and seems we more that in this to generate case,
why mind as new Y,
is just equal to lamber times x mind as new X,
we can also right that this the expectation of lambers,
we attens,
ex mind as new X,
Well, this we know is Landersquad. I'm the variance
of ex. So, we know that the Variance,
that Landersquid times the Variance of ex,
is the same thing as the variance of way.
So, he we have Landersquid times the Variance of
ex squid et. So, that's one factor,
Variance of ex, which we can, I solid,
we are left with Landersquad times there as of exqueent.
Sorry, it a level the squad times to there as
of exquad
Und das war eine Intition Effector Variance of X. But landesque
variance of excess equal two Variance of lining.
So we are rive at we of way,
variance of light times downs of excess,
is equal two lemn the speed variance of excess.
But and this is equal two that cover ands of ex and way squared.
So that we see in this degilered case,
the square converence is equal equal to the product of the two Variances,
ways in the non-dingenered cases, they sweared the square of the proviarities,
was small not then the product of the two Variances.
Both of these properties together, so equation three endequated one
on the previous slide, it's equation, here,
emply itsactly, what the cushy schwarz in here equality has.
Substated, then the cor Variance of ex employe squared,
is less then are equal to the Variance
of ex and the variance
of way
Okay. This is a proof which is little more publicated,
you should look at this proof,
which when you said, when it the back week,
but I can sure you that,
I would ask you to provide it,
proof as difficult as this one in eriton exam.
That's still, I think, it would be good if you understood it.
So let me post for a minute or give it just
deep portunity to raise your hand if you have any questions
relating to the material, I have out into these lecture.
Is please,
type in the question
Musik Buddy Elswurd like to pose
a question is also in curge to tabled or ready,
but I'm do waiting for the first question to come
Are you's dill type,
because I don't get any questions of far if it do
need time that please
raise you hand again
Ah, wer ist eine Question? Sie, could I please explain,
slide 30, and do these equations need to rememorized for the final exam.
It was do when about two questions coming in, slide 30,
I will just go back, not clear what a shell explain,
but a slide 30 with of me looking at Light 30,
and for the final exam, I mean,
I've just said, that I would ask before this particular proof,
or proof of similar difficulty in the pinel it and,
but I may ask, you to proof certain easier,
things, which we covered me.
I would provided was quite a view proofs or the cross
of the semester. So this not the case,
that you are dispensed of study in the proofs,
quite to the country, you should do this and you should be able,
if you, study know at an aster level,
at a master level where,
we educucate you to do independent research,
you should be able to proof
certain certain things
You don't need to memories, that in the sense of learning that by heart,
this absolutely not designed, this designed that you understand,
what you have to do in order to proof certain things,
and when you have understood this,
then these spleens follow more list magically proud from the other.
So therefore, it is mignistery to memories any equations,
what you should have in near head is what exactly to,
you want to proof, and what is the method of proving that,
where is I say, I will not ask you for complicated proofs,
or complicated difficult methods, but for easier stuff than this one,
are nee, ask you. I covered some approves
or ready in this lecture today,
which are ready just trivial proofs, these could very,
you easy to be from the final exam, actually typically,
I would, if I ask for a proof in the fan exam,
I would provly ask to something more difficult,
then let's say,
the easy decomposition of the Variance of random variable as the easy expectation of excreed minus the expectation squared.
So this would be an example of a proof,
I might ask you, but might ask you also some it.
Work difficult, and not as difficult is be a proof
of the pushy schwarz equa
Okay, now there are more are questions coming in and
I want for get topically and to go vectous land.
30, but let me first go to the questions,
but to relate to this slide here, why co-verriss of co-varence,
which we examplise eagual to call varence of ex eminders minders new ex,
why minders new why.
Well, simply because the co-Variance of ex and way is the thing thing as the expectation of
ex-minders new ex times why minders new us new way. I do,
you always compute the co-Variance and such away that you take the random variable,
minders it's expectation,
which would in this case be exmelled the expectation of ex-minders new ex, times why
minders new why. Now when you look at the random
variable ex-minders new ex, then this random variable of this random
variable obviously has an expectation of Zero, because you have subcracted the meme already.
And the same thing for why minders new way,
random very review why mind as new way has an expectation on Zero.
So when you compute the co-Variance of ex-minders new ex and why,
by nothing new why, then you have to subcract nothing,
you just tell to mind deply,
x-minders new ex with, why, when's new why,
which is the same thing,
as computing in
the co-variance of ex
and why
Ähm, And the nother question here this sentence please this allows,
ähm, das laus Ex Zero on a set of measure Zero.
Ähm, not quite clear, there is a similar be
used to which I don't quite and there stand, but like we just
go back in the slides to see see what you probably ment,
because I haven't than this oven, this thing with in measure Zero sets.
I don't move in the red, äh,
Ah, Dysillas X been different from Zero on a set of measure Zero. Yeah,
it's plain that, when I came to this slide,
I'm heavy to we explain it again,
the sit of measure or basically,
what would you have to know in order to understand that here?
And I would like to reificize that I won't ask about
these teams here and sets of measure Zero in the final league,
then because I don't cover this prototy in this link
job. The just for those of you will know about
it and it would probs otherwise take issue with my statement
that only a constant can have a varence of Zero,
which would not be true,
because of that some the random variables would have this step of property.
And then we those random variables, we are events,
happen our sit of measure Zero.
Now, what would you have to know is random variables
are actually mappings and basically factions from certain spaces too,
let's say,
the space of real members
And in the set on which the random variable is defined.
This is actually defined on a set of sets.
There maybe certain sets which have zero probibibility. So why there are not MT,
it is so unlikely that this particular event happens,
that the probability of this event sexually small and and any positive number,
and defore is this equal to zero.
Just purts to do try to explain this on an
intuitive basis. So post that the random variable,
which we talk about maps every single real number on the
number one
Meinst du,
you can basically think up it as a straight line,
horizontal to the ex access that the random verible ex would be straight line,
horizontal to the ex access at point one.
So with value one, for all real numbers,
that say with the exception of one real number.
So post for instance if we have on the ex access,
value zero at point to zero,
the value of the randomberry me is 10 rather thing one.
But for all other events, for all other excess,
the appropriate why Value would be one
In this case, the event that point Zero is drawn
from the set of a real members, has probibility Zero.
Because there are more than countably many real members.
And thinging our just one real member electors,
destember Zero, has a probibility of Zero.
Or more two atively preps, if you take your balken,
supposed it tasn't infinite lead short point, right,
to take your board pen and you just Kick,
it on the realize.
You just pick one number on the realize at random.
The probably that you hit exexactly the point zero is zero.
This just doesn't our current. Even if you try to hit the zero,
you will always, we're little bit billow or little bit of zero you
never meet exactly. Devery hit exactly point zero.
So the possibility that from the random draw at of the real numbers,
you just grown than brezero number zero.
So if Zero is maped on a number 10. And all
other real numbers are map on the number let's say, one.
Than the expected Value of this random variable is one.
And the variance of this random variable is Zero.
Because there is no probility for getting realization of the random
variable at level 10. Because just doesn't curred that this event
Zero will be chosen, brandomly, remember that I
told you yesterday.
I mean could pick
Ja, was denn? Hier. Hey, talk you steaday,
a continuous real it a continuous random verable takes on any real value mit Zero proviability.
So when we have a continuous random vaable,
the productability of peaking just one particular,
value or getting just one production a value in the random experiment,
is Zero, because this would trequire that the event,
which happens, is just one
particular event out of this set of more than countably many
sets in the different in the set defining that the random verable.
And this as as hey, as proviability of Zero
So, it is hards to explain this, that least from me. Patrick explain it any better,
then that with outgoing into the form and melkeys of definitions of sigma of random variables,
which would require to talk about sigma agibrast,
and this are sets of sets associated with certain proability measures.
I do not provide for this these lecture,
because these are somewhat demanding mathematical concepts,
which don't really help as much in understanding into magics,
which these lectures actually, about you will, if you persue,
econometrics on the doctoral level in the doctoral research,
which you going to the directure, you readical econometics,
you will have to learn this stuff.
But currently, when we board talk abouts the basic economics,
I think it is subvision to have then intuitive understanding, we know random variable as.
Something which happens at random. And then for is a orady said,
if you, at your stud, now,
or at least got sense of understanding for what I have been to say,
why measure zero set,
may have different Values from the rest of the events in this random variable,
so two to have measure one, then it is fine,
would be happy that is, you have tell me,
you you, I was do it at least persually,
but if you don't there resented it,
it all then don't out about it.
It it want we a part of the kind it them and it this
not accenture to understanding, anything else in the delecture.
I may the remark y just,
because I expects some of you to knownown about these things,
and yo would have have being techn delecture ture thato and
mayo to to thato to to to thato mas it and and
oto to to to and to to to to and to to to to it thato to to to and they as asero,
it it it to ceero,
it is and and and and rand rand and rand
rand the ment
ment merableable
Now with this, then we don't too page heardy,
to which the first question related, yes,
and there, I was as to explain this slide,
but I had explain this slide, I'm sorry,
can you please, a state your question,
more precisely,
and I think I have explain that as when I can,
the expectation of a linear combination of random variables is the linear combination of the expective Values of this randombergle.
Es is property of the expectation, other ranger,
and then the other things are just basically definitions.
So please, that we know more reprecisely,
what you would like it to explain about this.
And in the mean time, Somebody road reds me just
slight 335, which doesn't term we much?
Or did you want to say that I should do explain page side 35
run then skype 33? This is propsed the same doing to death that,
is this the case, please raise you hand,
you want be to explain page 35 and if you miss time,
when you's burke about the slight 30,
that the case
Polease raise your hand of this is the case.
A parently this is the case. Ja, okay,
yeah, but then I think at I explain that already.
So parently your question was the same pression,
the one which I have already a just patriot explain,
what this this remark in the brand to see,
by the way, mean,
that this the lost for explain different ponsy or our set of major see.
Okay, so I think we have seen it this and
this case every time to move on
Capilited the proof of the cuschwarz, inefolity, and you mean,
I ask why does that to explain the project schwarz in any inequality,
with his proof and the stetail where basically,
because I want to move to the coffee to completion,
which we can do only, if we have understood the cuschi schwarz, inequality.
Because the problems the form,
I introduce to or reintroduce you to the concept of it co-currence,
but the co- Variance is actually not really very informative,
because the converrence and take any value between
negative infinity inplus finitiv. So this difficult to
eterped and it doesn't make any sense to say,
the perveriance is great of the converiance is small.
This dependence completely on the units and which the variables are
measured. For instance, if one of my variables,
is measured in den Kilogramms, then a co-variance
maybe let's say of the order 10. But if I measure
the same variable in gramms than the co-variance,
movie in the order of 10.000, saying, so very.
So 10.000 has the 10 informational in content,
it is not possible to say,
at what properence is be your then we other it all depense on the united,
which the variables are easured.
And they for to say that converiance is great or small, is not sense.
It cannot be introperted this way. They only thing,
we can it fear, by looking at a converence etical variance is effect,
where they just negative, when the converence is negative,
or when this is positive. Even than,
we would still to see with a it is signalicantly negative,
or signalicantly positive, because they may, of course,
also be insignificant, co-variances,
but we come to testing and signalicance levels on
the little laisor
This is why I'm nose researchers,
don't talk offen about the cold variances, but rather,
the tried to norminize the cold variance, using the quitty schwarz, mequality.
And the result of this norminization is the cofficient of correlation,
which is certainly no, idefinite here again,
for any pair of random variables ex and why,
because ficient of correlation, which is typically the noted by,
role at here I index, it explicitly,
and with the reference to the two random variables, exemply.
The cofficient of correlation is defined as the cold variance between ex and why,
devided by the product of the standard deviations,
or exemploye.
So devided by Signal X individed basic my way
So this thing here is the co-ficient of correlation.
And he we can that make use of the curshi slots inequality,
because the co-ficient of correlation squared is then the co-variant squared,
devided but the Variance on X,
and the Variance of my.
And we know that the ratial of the co-variance squared,
you buyded by Variance of extent dividence of my,
is less then are equal to one,
by purtue of
the curshi
shots inequality
So we know that the squeed cofficient of correlation is always lesbinary to two one,
which emplies that the cofficient of correlation is always between negative one and class one.
So maybe equal, in the case of certain degeneral dependencies,
as we have seen from a proof of the cussy sparts in equality,
if it's not, genert, elivio dependency,
then the corfession of correlation will be greater than make it of one and smaller than one.
And this thing year is now non-grized,
because the cofficient of correlation is always in this iterall,
to we negative one add one.
So the cofficient of correlation measures, this drength of the
correlation between to variables ex it why. The closer,
the corfession of correlation is idea two one, or to negative one,
the more pronounced is the correlation between the variables ex and life.
And this is essentially the corre of economatures,
you may still recall this from a basic ecorder metrics,
if you have a simple, buy variot ecorder metrics,
then the die professionale, which you estime,
there is the confession of collision between the two variables.
Bright. So in the simple by very case and the regution estimets are exactly,
equal, two the confession of corvation, we cover this later when a speak about.
You know, mentrics that I show you this,
but in original prepare this recycling,
you to understand with the cofficient of coorrelation is,
it how it relates to the coverance and to the variance
of to the variancies of the two of the rebels to
the two randomberables ex and why
So, this is what researchers typically look at first,
when they look a data sense, prior to estimating inekind of models,
estimating inekind of regressions, typically researchers would look at their data,
and plot to variables or dance to other,
and check with that is correlation, which they maybe interested in.
And in particular of course, they are interested in correlations which are prononents,
so in compissions of correlation, which are other high,
idea lied close to one or close to native, one,
but that always are data as idealy structured as structured as we may possibly.
Wish, still, we would now that would be any exect post of leaner association between excit white,
if role of exquire, is epot 1,
and they would be and then exact,
negative linear association between except why if role is people to native when.
Which is typically attrus a case if the lean your association,
is really exect, usually this is not what we are interested in,
because nostly with made in error,
when we fight such an exactly a relationship,
in real world data, we usually have confessions of correlation,
which are great at the negative one, and's moder,
then when.
If there is no only your association between excite viet.
All, than we would have a profited of correlation would,
is closed to zero
So, the confession of collation gibt's als just as the standard.
Which we can judge the strength of any linear association between to variables.
That is already said, if this close to zero,
then the association is a purplely week and probably we shouldn't
base to strong in influencement such correlations.
We ass that is close to plus and negative a
one would imply, strong realization chips.
Gut, this concludes sectoral one point too other any
furthern questions
The Purpley not, go that surf with section one front,
three independence. Hobbly you know what independence is,
but I would define it, anyway,
to random variables ex of why are independent if and only if,
for all ex of why, it
is trues of all small ex,
and small why it is true,
that the joint density,
joint provility density function,
f off
Also FX Y, so the joint probility density function of
the randomary able ex and they randomary with why.
At this perpicular point, ex and why small ex and small why,
it's equal just to the product of the marginal densities.
All we introduced this concept, is that they
had the join density and the marginal density.
FX or ex is the marginal density of friend a verible ex,
if why, of why,
is the marginal density of random verible why
So, two randomarribles, exif what I said to be
independent if the joint probibility then city function,
is just the product of the reality marginal marginal PDS.
For all possible combinations of ex and way of small ex,
it's mall way. So, for all values of ex and way,
which ex and way can take on in the set of real numbers,
go a ranging from negative infinity,
to plus implitity for both small ex and small way.
This definition of independence here applies to both continuous and just
create randed rables and we would see very soon,
why this technical definition which at provide a coupher and dependents,
doesn't make sense auto on
intuitive basis
Now,
in the case of discreate from marables,
we can just re-right this independence condition as the probability that ex
is ecould to small ex and why is ecould small way,
is the same thing as the product of the proabilities that
ex esequal two small ex and pe and at why
is equal two small my
So for its is two toces of a drived two times roading addice.
We would usually usually say that the results of roading addice
far independent of each other. If the first roading addice that years,
the three, then that is no reason,
why the second role of the dice should to dealed a three,
two or shreed yeard any number related to three,
there's not the case,
the second role of the dice is independent of that and any number to we one and six,
can we roled on the second role with equal probablibility.
This is why, when we ply and probility the theory
to problems and volting roading addice, we tubically,
just multiply the probibilities of the first role,
with equal probibilities of the third role, of the same droad.
So that the probibibility of having say,
three in the first role and six in the second role,
would be the probility of having three in the first role
Times the probibility of having six and a second role.
So both of these would be 1.6. So what six times,
but six, would be one over 30-6.
So if the probibility of the joint event having three
and six as the result of two roles of dice,
that if this is one of a 36,
then we
would have independence
Trey to proof this that this property here,
is a same thing as this property here from the general definition of independence,
is a sent, the general definition of independence is true
both for this creed and for continuous, random veribles,
but for the discreed random very with you could equivitently,
right this condition in this way year.
The terms of notation we often right the independence as ex
is independent of why by using this kind of great pie simple turned upside out,
rand to you turn of the other way around than would be a great greek pie simple,
now it's just charged upside down
this means that ex is
independent of why
Now, the Implications are is follows and this proliced in
the evision for the definition for this technical definition, independent emplies that the co-verience is zero.
So if ex is independent of why, then this emplies that the co-verience between ex and why is zero. Why is this the cames,
well, very clearly, can see this,
if you have the joint tendencity of ex and why and the joint ensity is the product of the marginal endencities,
F ofX and FX of X and if wide of wide,
wenn das implies,
also that the expectation of extimes why of the random variables extimed wide,
is beaquel to the expectation of ex Times the expectation of why,
which then emplies that the coverages between the two is zero,
because usually will would know that the expectation of the product ex times why is equal to the expectation of ex and that times the expectation of why,
minus two times the coverages of ex and why.
And in this case if the two things are equal,
then because I eins turns mess is savery, equal to zero.
So independence of two randomaribles, emplies that the coverages of
ex and why is zero. And that ical variance of
ex and why zero meets that that is know relationship,
that lost least no linear relationship,
between ex and why,
which makes it at least suggestive that the veritals
maybe independent
Ein Forschnitt wie Implication just goes this way.
So, independence implized that the coverance is zero.
The reverse way is not true. So you know,
that two variables are have a co-variance of zero or and our uncorrelated, this doesn't
not yet emply independence. It does emply independence in the case
of nordely distributed variables that this is a special case,
for general distributional variables, it is not the case,
but the co-variance of two variables being zero emplies independence.
Only the reverse through independence implies coverance of Zero or correlation
of Zero. So, this basically the same thing,
if excess undependent of way,
then the correlation is zero between
the two variables
More over, independence emplies that the conditional mean equals be
unconditional. So if ex is independent of wide,
that the expectation of wide given ex this equal to the expectation of wide,
because verible ex doesn't is a not way helpful in expecting
variable wide sits the two are independent.
And so this will be the intuitive explanation of my this
is true
Yeah, and that have a really sense and prelated randomary with are not necessarily independent,
so dependence is stronger is a condition, then uncoordinatedness,
or prebellance of Zero, if we have to very able switch,
which have a corration of Zero,
we can not conclude at the two variables are independent.
They maybe sometimes of non-linear relationship between,
which we have encovered, it which we can't encovered,
in the linear type of analystics, between dependencies,
or the linear type of dependence andalysis,
which reply,
we look at role of ex
at one
Okay, I think, this is it for today,
are there any remaining questions and peace are raise your hand?
The parently not? Then I think you for a your tension,
please the remember to,
I know you don't have to right to student contribution this week,
I saw you're off for this week with his lincture from
next week on works you have to rattens,
student competition.
If not this week. So doctor, I've to remain
your that. Thanks for the attention and joy the rest
of the afterm