Linear Combinations of Independent Discrete and Normal Random Variables

Total payout $S = X_1 + \cdots + X_{50}$ for 50 independent policies requires either convolving distributions (exact but tedious) or using the normal closure property (fast). Knowing when to use each is a core Exam P skill.

If $X$ and $Y$ are independent discrete random variables, the PMF of $S = X + Y$:

$$p_S(s) = \sum_x p_X(x) \cdot p_Y(s-x)$$

For each target sum $s$, enumerate all valid $(x, s-x)$ pairs and sum the products.

HIGH-FREQUENCY: For small discrete distributions (dice, coins, small Poisson), direct convolution is fastest. List all valid pairs.

Some families are closed under independent summation:

• Bernoulli: $\sum_{i=1}^n X_i \sim \text{Binomial}(n, p)$ when $X_i \sim \text{Bernoulli}(p)$ iid
• Poisson: $\sum X_i \sim \text{Poisson}(\sum \lambda_i)$ for independent Poissons
• Normal: $\sum X_i \sim N(\sum \mu_i, \sum \sigma_i^2)$ for independent normals
• Gamma: $\sum X_i \sim \text{Gamma}(\sum \alpha_i, \theta)$, same scale $\theta$ required

Common mistakes

• Forgetting to square the coefficients in the variance. $\text{Var}(0.6X) = 0.36 \cdot \text{Var}(X)$, not $0.6 \cdot \text{Var}(X)$. Using 0.6 gives $\text{Var}(W) = 13.2$ instead of 7.20. Trap: 13.2.
• Subtracting variances for $X - Y$. Variance ADDS under independence regardless of sign. $\text{Var}(X-Y) = \text{Var}(X) + \text{Var}(Y) = 25$, not $16 - 9 = 7$. Trap: $\sqrt{7}$ as the SD.
• Using $\sigma^2$ instead of $\sigma$ when standardizing. The z-score divides by $\sigma$, not $\sigma^2$. Using 7.20 gives $z = 3.2/7.20 = 0.444$, wildly wrong. Trap: $\Phi(0.444)$.

Bottom line

• Convolution: $p_S(s) = \sum_x p_X(x) \cdot p_Y(s-x)$ gives the PMF of independent discrete sums
• Closure: Bernoulli sum = Binomial, Poisson sum = Poisson, Normal sum = Normal
• Normal linear combination: $aX + bY \sim N(a\mu_X + b\mu_Y, a^2\sigma_X^2 + b^2\sigma_Y^2)$
• Variance ADDS for differences: $\text{Var}(X-Y) = \text{Var}(X) + \text{Var}(Y)$ under independence

Exam shortcut

When you see "independent normal," immediately write the mean and variance of the linear combination. For the variance, draw a quick table: variable, coefficient, coefficient squared, variance, product. "Mean is linear, variance is quadratic." Mean of $aX + bY$ uses $a, b$. Variance uses $a^2, b^2$. "Variance never subtracts" (under independence). $\text{Var}(X-Y) = \text{Var}(X) + \text{Var}(Y)$, same as for the sum.

The full lesson (about 2,406 words, 16 min read) adds 3 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 3g

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