# Multivariate Random Variable - Numerical Characteristics with Python

March 25, 2018 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:

**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:

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:

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

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

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:

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.

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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.