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, 1)# and #N(7, 4)#. Take #n = 10000# for both variables. Next, create a variable #Y = 4X1 + 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 = 11#
The variance of #Y = 32#

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

X1 <- rnorm(10000, 1, 1)
X2 <- rnorm(10000, 7, 4)

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

Y <- 4*X1 + X2

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

#E[Y] = 4 \cdot E[X1] + E[X2]#
#Var[Y] = 4^2 \cdot Var[X1] + Var[X2] + 2 \cdot 4 \cdot Cov[X1,X2]#

You know that the respective means of #X1# and #X2# are #1# and #7#. In addition, you know that the standard deviations are #1# and #4#. 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 <- 4*1 + 7
VarY <- (4)^2*1^2 + 4^2 + 2*4*0

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