2.9 Combining Random Variables

Means of combined random variables

When combining random variables, means always add or subtract directly: E(X+Y) = E(X) + E(Y), and E(X−Y) = E(X) − E(Y). This holds true whether the variables are independent or not. For example, if the average cost of lunch is $8 and average cost of a movie ticket is $12, the expected total spent is $20.

Variances add — never subtract

If X and Y are independent random variables, variances always ADD, even when finding the difference: Var(X+Y) = Var(X) + Var(Y), and Var(X−Y) = Var(X) + Var(Y). This is because subtracting introduces more variability, not less. Standard deviation is then found by taking the square root of the combined variance.

Why independence matters

The variance addition rule only works if X and Y are independent — knowing the value of one variable gives no information about the other. If variables are dependent (correlated), this simple rule doesn't apply. Always check for independence before adding variances in AP Stats problems.

Key terms

E(X + Y)

E(X) + E(Y) — means always add, regardless of independence.

E(X − Y)

E(X) − E(Y) — means always subtract directly, regardless of independence.

Var(X + Y) for independent X, Y

Var(X) + Var(Y) — variances add.

Var(X − Y) for independent X, Y

Var(X) + Var(Y) — variances still add, never subtract, because combining variation increases uncertainty.

Requirement for variance addition rule

X and Y must be independent random variables.

Try an AP-style question

X and Y are independent with σ_X = 3 and σ_Y = 4. What is the SD of X + Y?

1. 5
2. 12
3. 7
4. 1