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

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

$\sigma_Y = |a| \sigma_X$

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.

Variance (adding): $\sigma_Z^2 = 8^2 + 6^2 = 64 + 36 = 100$

Standard deviation: $\sigma_Z = \sqrt{100} = 10$

For $W = X - Y$:

Mean: $\mu_W = 50 - 40 = 10$

Variance (variance adds for differences too): $\sigma_W^2 = 8^2 + 6^2 = 64 + 36 = 100$

Standard deviation: $\sigma_W = \sqrt{100} = 10$

Answer: For both $Z$ and $W$: mean is 90 and 10 respectively; standard deviation is 10 for both.

Variance of weighted sum: $\text{Var}(T) = (1)^2(0.5)^2 + (2)^2(0.7)^2 + (0.5)^2(0.4)^2$ $= 1(0.25) + 4(0.49) + 0.25(0.16)$ $= 0.25 + 1.96 + 0.04 = 2.25$

Standard deviation: $\sigma(T) = \sqrt{2.25} = 1.5$

Answer: $E(T) = 8.75$ defects and $\sigma(T) = 1.5$ defects.