Combining Random Variables


Suppose we have random variables X and Y.

If we add X and Y, we can form a new random variable Z.

The mean of Z is μZ = μX + μY

The variance of Z is σ2Z = σ2X + σ2Y

And if we subtract Y from X, we can form a new random variable we’ll call W.

The mean of W is μW = μX – μY

The variance of W is σ2W = σ2X + σ2Y

Notes on variance:

  • Whether we add or subtract two random variables, we still add their variances to find the variance of the newly formed random variable.
  • We can only add the variances if the two random variables are independent.
  • We can find the standard deviation of the new random variable by taking the square root of the variance of the new random variable.

Example 1

100 students take standardized science and math tests at the end of a school year. The table below shows the results of each test:


If we take the sum of a student’s science and math score, what would we expect the mean total score to be?

μsum = μS + μM

μsum = 82 + 85 = 167

We would expect the mean total score to be 167.

If we take the sum of a student’s science and math score, what would we expect the standard deviation of the total score to be?

Recall that we need the two variables to be independent in order to add their variances. In this case, we can’t assume that the science and math scores are independent so we are unable to find the variance, and thus the standard deviation.

Example 2

Researchers measured the heights (in inches) of 200 men and 200 women from a certain university. The summary statistics are shown in the table below:


What is the mean of the difference between the two heights?

μdiff = μM + μW

μdiff = 70 – 66 = 4

The mean of the difference between the two heights is 4 inches

What is the standard deviation of the difference between the two heights?

σ2diff = σ2M + σ2W

σ2diff = 32 + 22

σ2diff = 13

σdiff = √13

The standard deviation of the difference between the two heights is √13 inches

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