AP Statistics Unit 4 Notes: Random Variables, Expected Value, and Combining Outcomes

Random variable

A rule that assigns a numerical value to each outcome of a chance process.

Discrete random variable

A random variable that takes on a countable set of possible values (often integers like 0, 1, 2, …).

Outcome vs. random variable value

The outcome is the raw result of the chance process (e.g., HHTHT); the random variable value is the number computed from the outcome (e.g., 3 heads).

Probability distribution (discrete)

A list of every possible value of a discrete random variable and the probability that each value occurs.

Valid discrete probability distribution

A distribution where (1) each probability is between 0 and 1 and (2) all probabilities sum to 1.

Probability notation P(X = x)

The probability that the random variable X takes the specific value x.

Probability histogram

A graph with bars at each possible discrete value whose heights equal the corresponding probabilities; it represents a model (long-run pattern), not a dataset.

Cumulative probability

A probability of the form $P(X \le a$) (or similar) found by adding probabilities of all values meeting the condition.

Cumulative distribution function (CDF)

The function $F(x) = P(X \boldsymbol{\le} x),$ giving cumulative probabilities up to x.

Expected value

The mean of a random variable; the long-run average value over many repetitions of the chance process.

Mean of a discrete random variable

$μ_{X} = E(X) = \Sigma x_{i} p_{i},$ the probability-weighted average of the possible values.

Probability-weighted average

An average where each value is multiplied by its probability before summing; more likely outcomes count more.

Variance of a discrete random variable

$σ_{X}² = \Sigma (x_{i} - μ_{X})² p_{i},$ the long-run average of squared distance from the mean (weighted by probabilities).

Standard deviation of a random variable

σ_{X} = \text{√(σ_{X}²)}, describing the typical distance of X from its mean in the long run.

Variance–expectation shortcut

$σ_{X}² = E(X²) - (E(X))²,$ where $E(X²) = \Sigma x_{i}² p_{i}.$

E(X²)

The expected value of $X^{2}: E(X^{2}) = \Sigma x_{i}^{2} p_{i}.$

Fair game (expected value idea)

A game is “fair” when expected profit is 0 (fairness depends on expected value, not on winning half the time).

Profit random variable

If W is winnings and c is cost, profit can be defined as P = W − c, so E(P) = E(W) − c.

Linear transformation

A new variable formed by Y = a + bX (shift by a, scale by b).

Mean under linear transformation

If $Y = a + bX$, then $\mu_Y = a + b\mu_X$.

Standard deviation under linear transformation

If Y = a + bX, then $σ_{Y} = |b|σ_{X}$ (adding a constant does not change spread).

Additivity of expected value

For any random variables X and Y: $E(X + Y) = E(X) + E(Y)$ and $E(X - Y) = E(X) - E(Y)$ (does not require independence).

Independence (random variables)

X and Y are independent if knowing the value of one provides no information about the other; often from separate trials or independently selected individuals.

Variance of a sum/difference (independent case)

If X and Y are independent: $\text{Var}(X \boldsymbol{\pm} Y) = \text{Var}(X) + \text{Var}(Y),$ so \boldsymbol{\sigma}_{X \boldsymbol{\pm} Y} = \text{√(σ_{X}² + σ_{Y}²)}.

Linear combination (AP-level)

A form like T = a + bX + cY; if X and Y are independent, then $\text{Var}(T) = b²\text{Var}(X) + c²\text{Var}(Y)$ and $μ_{T} = a + b μ_{X} + c μ_{Y}.$