Discrete Random Variables

What Is a Random Variable?

A random variable $X$ assigns a numerical value to each outcome in a sample space.

A discrete random variable takes a countable number of values (often integers).

Probability Distribution

A probability distribution lists all possible values and their probabilities:

| $x_i$ | $x_1$ | $x_2$ | $\cdots$ | $x_k$ | |--------|--------|--------|-----------|--------| | $P(X = x_i)$ | $p_1$ | $p_2$ | $\cdots$ | $p_k$ |

Requirements:

• $0 \leq p_i \leq 1$ for all $i$
• $\sum p_i = 1$

Expected Value (Mean)

$\mu_X = E(X) = \sum x_i \cdot P(X = x_i)$

The expected value is the long-run average — what you'd expect on average over many repetitions.

Example: Roll a fair die. $E(X) = 1(\frac{1}{6}) + 2(\frac{1}{6}) + 3(\frac{1}{6}) + 4(\frac{1}{6}) + 5(\frac{1}{6}) + 6(\frac{1}{6}) = 3.5$

Variance and Standard Deviation

$\sigma_X^2 = \text{Var}(X) = \sum (x_i - \mu_X)^2 \cdot P(X = x_i)$

$\sigma_X = SD(X) = \sqrt{\text{Var}(X)}$

Rules for Transformations

If $Y = a + bX$:

• $E(Y) = a + b \cdot E(X)$
• $\text{Var}(Y) = b^2 \cdot \text{Var}(X)$
• $SD(Y) = |b| \cdot SD(X)$

Rules for Combining Random Variables

If $X$ and $Y$ are independent:

• $E(X + Y) = E(X) + E(Y)$ (always true, even if not independent)
• $E(X - Y) = E(X) - E(Y)$ (always true)
• $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$ (only if independent)
• $\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ (only if independent — ADD variances!)

Key Insight: Variances always add, even when subtracting random variables. This is because the variability of a difference is just as large as the variability of a sum.

AP Tip: The most common mistake is subtracting variances when computing $\text{Var}(X - Y)$. Always ADD variances!