Математика. Материалы курса

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We continue talking about the systems of random variables.

At the last session, we discussed how you can set the system of two random variables

At the end of the last session, we reviewed the definition of independent random variables.

Today we’ll understand what types of connection are possible between random variables, and to do this we introduce the concept of conditional distribution.

The first thing we look at is a conditional distribution series.

A conditional distribution series of a random variable under the condition  that the second random variable has a certain value is called conditional probability of the event ξ = x_i, provided that η = y_i.

It is defined by the formula of conditional probability.

And if you look there is number p_ij in the numerator, and in the denominator there is a private distribution of η, that is, it will be the sum of p_ij by j.

If the system is absolutely continuous, we can consider the conditional distribution function and the conditional distribution density.

First we introduce the conditional distribution function.

Suppose a random variable η accepted value y₀.

It is necessary to find the distribution of the random variable ξ provided that η = y₀.

The distribution function is set as follows.

Let us first see a certain period, not point y₀, (we know that there is a problem when a specific point is taken), but the interval from y₀ to y₀ + Δy.

Let Δy be reduced.

Then consider an event that the random variable took some value less than x, provided that the second random variable is here in this period.

Then we can set the distribution function, which is under the limit.

Then we let Δy to zero and in the limit we obtain the distribution function at point x, provided that η = y₀.

If such a limit exists, then we can speak of a conditional distribution function.

Further, if we calculate the distribution function, we note that it is equal to the integral from minus infinity to x of the joint distribution density, divided by the private distribution density of value η.

Differentiate both sides by parameter x.

On the left there will be the distribution function, on the right in the denominator we have a specific number, but in the numerator an integral with the upper variable limit.

We get the values for the conditional distribution density, which is shown on the slide (see. In the video).

That is, this equation allows us to calculate easily the conditional distribution density.

An important concept that will to be used further in mathematical statistics, is conditional expectation.

Conditional expectation  is expectation of value ξ provided that η = y₀.

We note that this expectation depends on what is the value of random variable η, and it will be denoted by ф (y₀), i.e. some function.

And it is a scalar function, not a random one.

A dependence of the mean value ξ of n we call regression of ξ to η.

If we consider the expectation of a random variable η, provided that value of ξ is x_0, this will be regression η to ξ.

You shouldn’t be confused here.

The graph of function ф (y₀) will be called the regression curve.

Here we pause and say that if ф is constant, then we can say that no matter what is the value of n, the mean value of ξ is unchanged.

Such a dependence is called correlation.

The slide shows three types of dependence.

The strictest dependence is functional, when the value of a random variable can give the value of the second random variable.

The least strict dependence is statistical when there is no independence, i.e., the value of a random variable affects the distribution of the second random variable.

The third dependence is discussed here, this is our new correlation dependence.

This is the dependence of the mean value of a random variable from the value of the second random variable.

In order to examine the correlation dependence let’s look at a few more definitions.

The first parameter is a numerical characteristic of random variables, this is covariance.

It is denoted cov(ξ, η) and characterizes the linear dependence of two random variables.

Covariance is considered the expectation of the product of two centered random variables.

Let me remind you that the random variable is centered if its expectation  is subtracted  from it, its middle will move to zero.

Then we note that the main feature of the covariance is that it is commutative.

By definition, it is clear that it does not matter in which order we consider the random variables.

Yet another convenient formula for calculating covariance is given on the slide with its derivation.

Just as we looked for a convenient formula of dispersion, there is a more convenient formula for covariance.

Once again I say, that this formula makes sense to use only for manual calculations.

That is, if you calculate with your computer, you do not need to use this formula because the formula by definition calculates faster, since you do not have the problem of the accuracy or complexity of calculations, as strict as by hands, I would say, it is possible to calculate covariance by definition.

Properties of covariance.

What other properties can be considered?

It is important to look at the sign of covariance.

Let's turn our attention to the fact that this is some expectation.

If dispersion is the expectation of the a square, then the expectation of the product of two different centered random variables can be either negative or positive.

Thus, depending on the sign we are talking about the presence of positive or negative dependence between random variables.

That is, if I have positive covariance, it means that with an increase in the value of a random variable, the second also  increases in average, and vice versa, if the covariance is negative, with an increase in the value of a random variable, the second random variable  decreases in average.

Do not forget about the word "in average".

The second point.

Covariance is bounded above by the root of the dispersions product.

We will not prove this.

You can either carefully write down, or use the systems and functions of two random variables that are not included in this course.

Anyone interested in that can look into the subject of functions of two random variables in any textbook and see the proof of this fact.

It is important that if random variables are independent, then they will also be uncorrelated, i.e., if the covariance is zero, we say that the values are uncorrelated.

This means that non-dependence determine non-correlation, but not vice versa.

If we recall the picture with the types of dependencies (go a few slides back), you will see that if the point is out of statistical dependence, it is necessarily outside the correlation one.

The converse is not true.

So we see in the picture, there are a set of pairs where there is a statistical dependence, and there is no correlation one.

Let us prove it.

That it is not really empty.

And to do this just give an example, when there is a statistical dependdence, and there is no correlation one.

In practice, we count the particular distribution for this example, which is shown on the slide.

Here we have a pair of random variables uniformly distributed in the circlw with radius 2.

When we say, uniformly distributed, we mean that the probability of each point is exactly the same.

So, like in the one-dimensional case, in the two-dimensional case the distribution density is constant.

In this case, since the probability of hitting a point in any point of this area is 1, that is, we know that the normalization condition works, and we get the area under the distribution density – this is a cylinder, we know that the volume of the cylinder is its height by its base, then to find the height – this is the  value of distribution density, - it is enough to calculate the base area and divide on e by the base area.

That, in our case will be one by 4 π, because the area of the circle of radius 2 is equal to 4 π.

We, therefore, can find private distribution for the random variables ξ and η.

The product of these partial distributions will not be equal to the joint density, that is, it will not be constant on the circle and zero outside this circle.

Therefore, in this case, the valueы will be dependent, that is, they are not independent.

Let us prove that in this case they will not be correlated.

Since we have aт absolutely symmetrical picture, then, whatever value of x I take, the distribution of y is uniformly symmetric with respect to the origin, and therefore, the mean value will be zero.

That is, no matter what value of x I take the mean value of y will be equal to zero.

This means that the expectation will always be equal to one and the same value.

This means that the values are uncorrelated.

You can check that in this case, the covariance is zero.

Check it out yourself.

Thus, if the random variables are independent, they will necessarily be uncorrelated.

The converse is not true.

Covariance is inconvenient in that if the variance is large enough, then covariance may be large.

That is, if we consider random variables in different units, the result is covariance will take on different values.

Thus, using covariance, we cannot compare the dependence between different pairs of random variables.

Where is a stronger connection, where is less strong connection?

All depends on the measuring units.

This is not good.

So it makes sense to normalize the correlation.

Normalized covariance is called the correlation coefficient, that is, we know that covariance is not greater than the square root of the dispersion product, so covariance is just divided by this root and get the correlation coefficient.

Again, the correlation coefficient in absolute value does not exceed the square root of the variances product by the square root of the variances product, i.e. one.

This may take both negative and positive value.

If the correlation coefficient is +1 or -1, then we can say that there is a functional dependence between the variables.

At the same time, it is important that it is linear, that is, in practice, the linear dependence is most common.

This is what we'll talk about again in mathematical statistics, when we talk about dependence

And the second property.

If random variables are independent, the correlation coefficient is equal to zero.

The converse is not true, we have already shown, that is, covariance is not zero, hence, the correlation coefficient is not 0 either.

Here the lecture is over.

Good luck.