4.9 Combining Random Variables

Part of Probability, Random Variables, and Probability Distributions in AP® Statistics. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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Sum and Difference of Random Variables

For random variables ${X}$ and ${Y}$ :

Means always add/subtract:

\mu_{X+Y} = \mu_X + \mu_Y

\mu_{X-Y} = \mu_X - \mu_Y

Variances add (only if X and Y are independent):

\sigma^2_{X+Y} = \sigma^2_X + \sigma^2_Y

\sigma^2_{X-Y} = \sigma^2_X + \sigma^2_Y

Note: For the DIFFERENCE ${X - Y}$ , the variances still ADD (not subtract).

Standard deviations:

\sigma_{X \pm Y} = \sqrt{\sigma^2_X + \sigma^2_Y} \quad \text{(if independent)}

Why do variances add for differences? Think of it this way: if you're measuring the distance between two independently varying points, uncertainty from both adds up.

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