Probabilities and moments such as the mean, variance of joint random variables is not an unknown topic for the reader. Calculating the expected value of two independent variables is a linear combination of them. For instance let’s take \(u(X,Y) = X + Y\), and let’s say we want to find the mean of \(u\). We already know, for previous studies that this can be written:

$$ E[u(X,Y)] = E[X+Y] = E[X] + E[Y]$$

Starting to remember? This type of calculations we are used to doing. In case of probabilities, we know that these are calculated the same way as the mean, we define a region \([0,p_x]\cdot [0,p_y]\) and integrate over it with the linear combination of these two functions, let’s say

$$ \int \int_{[0,p]\cdot [0,p]} u(x,y)f(x,y)dxdy.$$

Mostly for linear combinations, moments will be the greatest interest but nonetheless one must know how to find certain probabilities in case it is needed.

**Learning Outcome**

**Topic 3.h: Multivariate Random Variables – Calculate probabilities and moments for linear combinations of independent random variables.**