Probability theory: Multivariate random variable, numerical characteristics.

This is part of the course “Probability Theory and Statistics for Programmers”.

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Probability Theory For Programmers

The numerical characteristics of the random variable were already introduced in one of the articles. Now we will look at moments of a vector of random variables.

The initial moment of the k, s order of a system (X,Y) is expected value of multiplication of X in k-th degree and Y in s-th degree:

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the initial moment

The central moment of the k, s order of a vector (X,Y) is expected value of multiplication of X central in k-th degree and Y central in s-th degree:

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the central moment

In order to calculate those moments for a vector of discrete random variables by knowing their distribution low and expected value we can use formulas:

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initial and central moments for discrete

In order to calculate moments for a vector of continuous random variables by knowing probability density and expected values we can use formulas:

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initial and central moments for continuous

The first initial moments represent the expected values of random variable X and Y of the vector:

expected values

The combination of expected values is characteristic of the position of the vector. Geometrically, these are the coordinates of the midpoint for the plane around which scattered the point (X, Y).

The second initial moments represent variances of random variable X and Y of the vector:

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variances

A combination of variances characterizes the scattering of a random variable in the direction of the axes.

Let’s take a look at the example. We have the vector (X, Y) of discrete random variables and distribution low. What is the expected value and variance of the vector? For this example, we will use the “homemade” function which will return probability for given x and y.

In this example, probabilities go up with values of x and y. I show this on the plot by increasing the size and opacity of points according to probabilities. The red point is the combination of expected values of the vector and the black one is the combination of variances.

Now is the time to look at specific for the vector of random variables characteristic — correlation moment.

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correlation moment

Correlation moment describes not only scattering of variables X and Y, but also their relations. If the correlation moment of two random variables is different from zero, this is a sign of the existence of a relation between them. If we want to look only at the characteristic of relations between variables we need to use different characteristic — correlation coefficient, since if one of the variables very close to expected value the correlation moment will be very low, no matter how close dependence.

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correlation coefficient

Let’s find a correlation coefficient for the previous example.

When r < 0 we have a negative correlation and if r > 0 — positive. A positive correlation between random variables means that as one of them increases, the other tends to decrease on average and negative means that as one of the random variable increases, the other tends to decrease on average. As you can see from the example we have a negative correlation.

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