Good day.
So we will talk about probabilities.
And integration with respect to probability distributions.
First of all, I'd like to remind you.
For what the probability distribution is?
If we have.
Random variable X.
Well then.
Probability distribution is the image of measure being.
So measure B is on this space.
Omega
a random variable.
Is.
Real valued.
And for any Borel set.
On the vertical line.
They.
Measure PX of B is simply the original probability measure of X -- 1 of beer.
This is X minus around of B and of course as soon as we're talking about.
Random variables, so we have a fixed probability space Omega air.
P and of course a random variable is a measurable mapping from Omega.
Do real numbers.
So this is just to remind you.
But revision if you like.
No theorem.
On page 27.
Says the following. If we have probability space if we have random variable.
X, which is measurable mapping as you know.
But then.
One can introduce the probability distribution of X.
And after that, for a Borel measurable function G.
So the integral Geo fixer for Mega GPO for mega.
Equals integral over real numbers.
GFX or weather. Expect to the probability distribution.
If you remember your calculus, which is very important as I said many times.
But this formula represents just change your variable.
So we pass from Omega.
2X.
And they both integrals are well defined or not simultaneously.
Here.
Deep this integral.
And this integral are introduced in the sense of Libya.
Your function G.
Is for example.
Continuous in case X or form agrees continuous and then we have just normal Riemann integrals.
And your background in calculus comes.
In this stage.
So suppose so. We have 4 mega standard.
Segment 01
Bottle Sigma Field and really big measure.
Repose access
increasing function. Suppose it is continuous.
And the.
Universe function or my golf X is differentiable or then this is the standard.
Formula from calculus.
Change of variable after we pass from Omega to X.
But the limits of course change.
Instead of 0 or we have X of 0. Instead of wonder we have extra one and don't forget about the derivative of the inverse function.
Now, if you compare this formula with the previous one.
Will we see that?
Well, this.
Term derivative of the universe function DX.
Coincides with.
Deep PX.
In the following sense, for any set a.
Bottle measurable of course, but the value of the.
Measure PX probability.
PX of A equals integral over a.
Off of this.
Universe function derivative.
Of course.
Multiplied by the indicator.
Becausw
becausw they arrange.
Of X is.
From X0 to X441.
Again, this can be.
Illustrated on the.
Picture.
OK here is.
Increasing function.
01
and just change your variable.
But they arrange your faxes.
EXO 0.
And.
Extra for one.
For the integration is.
Only.
Over this
area.
One day, but real numbers. So that's the way indicator appears.
From Microsoft 0 to X1G X derivative for the inverse function. This is standard calculus, your background.
Is serious, so for examples.
Simplest random variable is a constant, or we calculated, or the probability distribution that is just direct measure at point A.
And for every function G.
Borel measurable.
Or more generally, can be arbitrary function integral with respect to.
With respect to.
Initial.
Probability measure.
On Omega equals G of A&E equals integral of geofacts.
With respect to the Dirac measure at point A.
Oh, this is the real example. Maybe it.
More general case is about discrete random variable, which takes values AI with probabilities Pi.
No then.
The probability distribution is just combination of direct measures at corresponding points with coefficients Pi.
And in this case.
Integral of Geofacts with respect to initial probability measure.
Equals.
Integral over real numbers.
With respect to the probability distribution.
Equals.
Some of the values of the function at the points AI.
With the corresponding coefficients Pi. So this is combination of direct measures, so every discrete random variable has probability distribution or which is combination.
Off
Derek measures.
Let the corresponding points.
Second example.
Oh, here we have.
Random variable of this form.
And the graph.
Looks as follows.
X equals.
4.
In the when the middle part of Omega.
And it increases.
With coefficient 5 and decreases on the left and on the right.
So here is a 1/3.
It is too.
Or three.
So what is that?
Probability distribution.
Now the formula is on the screen.
And I'd like to clarify it.
So if we have, of course if.
.4 belongs to be then.
For this full interval belongs to the preimage and.
Coefficient of 1/3 appears becausw 1/3.
Becausw
for 1/3.
Is the measure of this interval.
From 1/3 to up to 2 / 3.
So if.
B's apart.
Of this form.
AB.
On the interval from zero to 5 / 3.
So if we have.
Intersection of be with.
Interval zero 5 / 3.
It is here.
But then.
The.
Complete pre image.
Or will have.
For real well in this simple case, or when we have interval or we have.
A.
Forward 5 here and be over 5 there.
And the same story.
There.
So are they.
Initial measure, which is lebec measure.
Is applied for this interval.
Hey.
Or were five B 05? So this is.
For 1/5 of the Libyan measure.
Off
baby.
And more generally, if we have an arbitrary set be on the vertical.
Lion.
Well then the complete pre image.
Will have two parts.
And the measure of the complete pre image will be 1/5.
Off
with the.
1/5 of the Libya measure of this intersection.
So look at this picture to understand.
But the calculations better.
So interval AB transforms to the interval.
Abo 5 A 05 B 05.
And the complete pre image is a beer over 5 and the measure of the company preimages 1/5.
And we have two parts of this.
Skype.
On the left and on the right.
So this is explanation to this formula.
And this can be buried them down as one expression of this shape 1/3.
Both for the Dirac measure and 2 / 5.
Off the Libya conversion.
Oh
integral would be.
With respect to the Libya merger, again, indicator a peers in the.
Integral becausw
the cause.
Here we have.
Interval
from zero.
Store.
5 / 3
All other.
Will you Sunday.
Video numbers.
But not needed.
So for a Riemann integrable function G.
Integral GGP in concise with integral over a real numbers with respect to the probability distribution.
This is illustration to the theorem.
And that equals 1/3 G of four.
Plus integral.
From zero to 5 / 3 function geofacts 2 / 5 G of X.
Well, this is second example illustrating the theorem.
Next example.
We have
Omega 0 or 1/2 open.
In federal.
Borrow.
Measurable Sigma field.
Bands of the early big measure.
Function random variable X.
Fish are this form.
So 01.
Her open interval.
And.
So we have a logarithm which goes to Infinity.
This is increasing function.
Continuous of course. The inverse function.
Is on the screen or 1 -- 8 -- X.
For positive value suffix.
And again, indicator of positive numbers appears.
And they.
Probability distribution of any borell set A equals integral over a.
Offer the.
Off the.
Difference derivative of the inverse function.
So the Omega.
GX?
Equals U2 -- X.
So remember, if the function X was similar but decreasing.
In this Formula One has to put absolute value of the derivative.
So here we have lucky that derivative is not negative becausw probability distribution.
Cannot be negative.
Or integrate over a.
Positive function and in case of the inverse function is decreasing with take the absolute value.
So.
To be more precise, one has to write absolute value.
So in particular.
If we are interested in the cumulative distribution function, probability of that random variable is smaller than T.
Then we have.
The X or the interval minus Infinity.
And if you integrate this function.
All were infinite interval.
The result will be.
O of course, for negative values of T.
And the one minus exponent for positive values of T. And you must recognize here for the exponential.
Random variable.
Alright, we had many examples students like examples ask for.
More and more examples will continue. Consider many examples, I promise.
Thank you for your questions goodbye.