Combining Random Variables

Unit 4 · Probability, Random Variables & Distributions

What AP Stats asks here

Two rules govern combinations of random variables: means add with the obvious sign, and variances add (not SDs) for independent variables — even when you are subtracting. The trap is treating SDs as if they add directly. They do not, because variance is a squared quantity and partial cancellation of independent fluctuations shrinks the combined spread.

Means and variances for independent X, Y

Means add (with sign)

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

Variances always add

\sigma_{X \pm Y}^2 = \sigma_X^2 + \sigma_Y^2

SD comes from variance

\sigma_{X \pm Y} = \sqrt{\sigma_X^2 + \sigma_Y^2}

Sum of n iid RVs

Mean

\mu_T = n\mu

Standard deviation

\sigma_T = \sigma \sqrt{n}

AP Tip: Variance is a squared quantity. The SD addition rule is sqrt(a² + b²), not a + b. Picture a right triangle — the hypotenuse is always less than the sum of the two legs.

Caution: On subtraction the means subtract but the variances still add. The SD of X − Y equals the SD of X + Y for independent variables — a surprising but correct result.

Type 1

Means add (with sign)

$\mu_{X+Y} = \mu_X + \mu_Y$ and $\mu_{X-Y} = \mu_X - \mu_Y$ — always, even when X and Y are correlated.

Example 1

A student's math SAT has

\mu_M = 600

and verbal

\mu_V = 580

. If we treat them as independent, what is the mean of the total

T = M + V

? (A) 590 (B) 1000 (C) 1180 (D) Cannot tell from the means alone.

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Type 2

Variances add for independent RVs

For independent variables, $\sigma^2$ of the sum equals the sum of the variances. The same rule holds when subtracting because Var(−Y) = Var(Y).

Example 2

X

and

Y

are independent with

\sigma_X = 15

and

\sigma_Y = 12

. What is

\sigma_{X-Y}

? (A)

15 - 12 = 3

(B)

\sqrt{15 - 12} \approx 1.73

(C)

\sqrt{15^2 + 12^2} \approx 19.21

(D)

\sqrt{15^2 - 12^2} \approx 9

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Type 3

Standard deviations do not add

Combine variances first, then take the square root. Adding SDs overstates the combined spread.

Example 3

A student writes

\sigma_{X+Y} = \sigma_X + \sigma_Y

for independent

X, Y

with

\sigma_X = \sigma_Y = 1

. Which critique is correct? (A) The student is correct — both SDs are positive so they add. (B) The student gets 2, but the actual independent-sum SD is

\sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.41

. Combine variances, then take the square root. (C) The rule works for sums but not differences. (D) The rule is fine because the variances cancel.

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