Variance and Standard Deviation for Conditional and Marginal Distributions

The reinsurer needs the overall variance of total claims. That depends on both randomness within each segment and differences between segments. Eve's law decomposes the total into within-group and between-group components, and the between-group piece can dominate.

The conditional variance of $Y$ given $X=x$ measures spread within the subpopulation where $X=x$:

$$\text{Var}(Y|X=x)=E[Y^2|X=x]-(E[Y|X=x]{)}^2$$

The conditional variance depends on $x$. Different values generally produce different variances. When viewed as a function of the random variable $X$, $\text{Var}(Y|X)$ is itself a random variable.

HIGH-FREQUENCY: Exam problems give conditional means and variances for several groups, plus mixing weights, and ask for the overall SD.

$$\text{Var}(Y)=\underset{\text{within-group}}{\underbrace{E[\text{Var}(Y|X)]}}+\underset{\text{between-group}}{\underbrace{\text{Var}(E[Y|X])}}$$

Term 1 (within-group): Weighted average of conditional variances:

$$E[\text{Var}(Y|X)]=\sum _x\text{Var}(Y|X=x)\cdot p_X(x)$$

Term 2 (between-group): Variance of the conditional means:

$$\text{Var}(E[Y|X])=\sum _x(E[Y|X=x]{)}^2{p}_X(x)-(E[Y]{)}^2$$

Common mistakes

• Adding conditional SDs instead of variances. The within-group SD is NOT $\sqrt{3}(1/2)+\sqrt{8}(1/3)+\sqrt{20}(1/6)\approx 2.554$. Eve's law operates on variances, not SDs. Trap: 2.554 as the "within-group SD."
• Forgetting to square the conditional means. Computing $E[Y]=13/3$ and then reporting $(13/3{)}^2=169/9$ as the between-group variance confuses the second moment with the variance. You need $E[(E[Y|X]{)}^2]-(E[Y]{)}^2$. Trap: 169/9 as the between-group variance.
• Using equal weights when segment proportions differ. Equal weights 1/3 each give $E[Y]=(2+5+10)/3=17/3$ instead of 13/3. Trap: 17/3.

Bottom line

• Eve's law: $\text{Var}(Y)=E[\text{Var}(Y|X)]+\text{Var}(E[Y|X])$, within-group plus between-group
• Conditional variance $\text{Var}(Y|X=x)$ uses the conditional PMF and measures within-group spread, generally varying with $x$
• Variances add, SDs do NOT: always decompose variances, then take the square root only at the end
• If all conditional means equal: the between-group term vanishes

Exam shortcut

Set up a three-column table: weight, $E[Y|X=x]$, $\text{Var}(Y|X=x)$. Compute four quantities in order: (1) $E[Y]$, (2) within-group, (3) $E[(E[Y|X]{)}^2]$, (4) between-group = item 3 minus item 1 squared. Add items 2 and 4. "Variances add, SDs don't." Eve's law adds variance components. NEVER add or average standard deviations. "Mean, Within, Between, Total": the four-step computation order.

The full lesson (about 2,706 words, 18 min read) adds 3 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 3d

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