Combinations of random variables

4. Probability Distributions: Practical 4

Combinations of random variables

In the theorical part of this course, you learned the following equations for combining random variables:

If #X = aY + bZ# then:

#E[X] = a~E[Y] + b~E[Z]#
#Var[X] = a^2Var[Y] + b^2Var[Z] + 2ab~Cov[Y,Z]#

If #X = aY - bZ# then:

#E[X] = a~E[Y] - b~E[Z]#
#Var[X] = a^2Var[Y] + b^2Var[Z] - 2ab~Cov[Y,Z]#

Generate two random variables #X1# and #X2# from two different normal distribution #N(1, 3)# and #N(6, 1)#. Take #n = 10000# for both variables. Next, create a variable #Y = -6X1 + X2#.

Calculate the mean and variance for the combined variable #Y# by applying the theoretical equations for combining random variables given above and the values for the population parameters of #X1# and #X2#.

The mean of #Y = 0#
The variance of #Y = 325#

First, you should create the random variables #X1# and #X2# with the function rnorm(n, mean, sd).

X1 <- rnorm(10000, 1, 3)
X2 <- rnorm(10000, 6, 1)

Now, combine these following the function as described in the question.

Y <- -6*X1 + X2

Following the equation for the combination of random variables, the mean (expected value) and variance can be calculated as follows:

#E[Y] = -6 \cdot E[X1] + E[X2]#
#Var[Y] = -6^2 \cdot Var[X1] + Var[X2] + 2 \cdot -6 \cdot Cov[X1,X2]#

You know that the respective means of #X1# and #X2# are #1# and #6#. In addition, you know that the standard deviations are #3# and #1#. Recall furthermore that the variance is the squared standard deviation. The variables #X1# and #X2# are independent.

Therefore, you can calculate the mean and variance of #Y# as follows:

EY <- -6*1 + 6
VarY <- (-6)^2*3^2 + 1^2 + 2*-6*0

New example