➕ Combining Random Variables (Sums and Differences)

Often in statistics, we work with linear combinations of random variables. For example, total profit might be the sum of profits from two stores, or net gain might be a difference. Understanding how means and variances combine under these operations is critical for inference and prediction.

Linear Transformations: $aX + b$

For a random variable $X$ with mean $\mu_X$ and standard deviation $\sigma_X$, any linear transformation $Y = aX + b$ has:

$E(Y) = E(aX + b) = aE(X) + b = a\mu_X + b$

$\text{Var}(Y) = \text{Var}(aX + b) = a^2 \text{Var}(X) = a^2 \sigma_X^2$

$\sigma_Y = |a| \sigma_X$

Key insight: Adding/subtracting a constant shifts the mean but does NOT change the variance or standard deviation. Multiplying by a constant $a$ scales both mean and variability; the standard deviation scales by $|a|$.

Sums and Differences of Two Independent Variables

For independent random variables $X$ and $Y$:

$E(X + Y) = E(X) + E(Y) = \mu_X + \mu_Y$

$E(X - Y) = E(X) - E(Y) = \mu_X - \mu_Y$

$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) = \sigma_X^2 + \sigma_Y^2$

$\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y) = \sigma_X^2 + \sigma_Y^2$

Note: Variance adds for both sums and differences (the variance of $X - Y$ is the same as for $X + Y$).

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

Weighted Sums and Multiple Variables

For independent variables $X_1, X_2, \ldots, X_n$:

$E(c_1 X_1 + c_2 X_2 + \cdots + c_n X_n) = c_1 \mu_1 + c_2 \mu_2 + \cdots + c_n \mu_n$

$\text{Var}(c_1 X_1 + c_2 X_2 + \cdots + c_n X_n) = c_1^2 \sigma_1^2 + c_2^2 \sigma_2^2 + \cdots + c_n^2 \sigma_n^2$

Conditions for Independence

These rules require independence. If variables are dependent (e.g., high values of $X$ tend to occur with high values of $Y$), the variance formula must account for covariance, which is beyond the AP Statistics scope.

Worked Example 1: Linear Transformation

A store's daily revenue is $R = 50P + 200$, where $P$ is the number of items sold. Suppose $E(P) = 100$ and $\sigma(P) = 20$.

Find $E(R)$ and $\sigma(R)$:

$E(R) = 50 \cdot 100 + 200 = 5000 + 200 = 5200$

$\sigma(R) = |50| \cdot 20 = 1000$

The mean revenue is $5200 per day, with standard deviation of $1000. Multiplying by 50 scaled the variability proportionally.

Worked Example 2: Sum of Independent Variables

Two independent vending machines have daily revenues: Machine 1 with $\mu_1 = 100$, $\sigma_1 = 15$; Machine 2 with $\mu_2 = 120$, $\sigma_2 = 20$. Let $T = R_1 + R_2$ be the total.

Mean of sum: $E(T) = 100 + 120 = 220$

Variance of sum: $\text{Var}(T) = 15^2 + 20^2 = 225 + 400 = 625$

Standard deviation of sum: $\sigma(T) = \sqrt{625} = 25$

Even though Machine 2 alone is more variable ($\sigma_2 = 20$ vs. $\sigma_1 = 15$), the combined standard deviation is smaller than the sum of individual SDs ($15 + 20 = 35$) because variance adds, not standard deviation.

Common Pitfalls

⚠️ Variances Add, Not Standard Deviations: A common mistake is to add standard deviations: $\sigma_{X+Y} \neq \sigma_X + \sigma_Y$. Instead, $\sigma_{X+Y} = \sqrt{\sigma_X^2 + \sigma_Y^2}$, which is always less than the simple sum.

⚠️ Variance of Difference = Variance of Sum: Do not forget that $\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$, not subtraction. Negative values don't reduce variance.

⚠️ Independence Assumption: These rules assume independence. If $X$ and $Y$ are correlated, the formulas are invalid. Always verify or state the independence assumption.

Calculator Tip

💡 TI-84 / TI-Nspire: To work with combined variables, define mean and variance for each component, then use the rules above manually (e.g., $E(2X + 3Y) = 2 \times E(X) + 3 \times E(Y)$). There is no built-in function; apply the formulas by hand and verify your arithmetic.