Calculate probabilities and moments for linear combinations of independent random variables

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.


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