AP Statistics Class 11: Random Variables & Expected Value — 12 SOMATH Posters, Theory & 60 Practice Questions

Definition

A random variable is a numerical value determined by the outcome of a random process. We usually write it with a capital letter — X, Y, Z — and its individual values with lowercase letters.

Example 1: Roll a fair six-sided die

Let X = the number rolled. The possible values are 1, 2, 3, 4, 5, 6, each equally likely with probability 1/6.

Example 2: Flip 3 coins

Let X = the number of heads. The possible values are 0, 1, 2, 3. Even though the outcome of each flip is unknown, X is a well-defined number every time we do the experiment.

Why we call it "random"

The value depends on chance. Before the experiment we cannot know what X will be — only what values it could take and how likely each value is.

Why it matters

Random variables let us apply arithmetic — sums, averages, comparisons — to random processes. Every AP Statistics topic that follows in this unit (probability distributions, expected value, variance, binomial, geometric, sampling distributions) starts with a random variable.

Two kinds of random variables

Every random variable is either discrete or continuous. Which kind it is decides how we write its probability model and how we compute probabilities.

Discrete — you can count

A discrete random variable takes countable values (finite or countably infinite). The possible values can be listed: 0, 1, 2, 3, ….

• Number on a die: 1, 2, 3, 4, 5, 6
• Number of heads in 3 coin flips: 0, 1, 2, 3
• Number of students absent: 0, 1, 2, …
• Number of goals in a soccer game: 0, 1, 2, …

Continuous — you measure

A continuous random variable can take any value in an interval. The possible values form a continuum — infinitely many values, and between any two you can always find another.

• Height of a person: 150.5 cm, 150.6 cm, 150.63 cm, …
• Time to complete a test: 23.4 minutes
• Temperature: 72.3°F
• Weight of a package: 1.27 kg

Bar chart vs density curve

Discrete distributions are shown with a bar chart (or histogram): one bar per possible value, height = probability. Continuous distributions are shown with a density curve: probability is area under the curve, not the height at a single point.

What a probability distribution shows

A probability distribution of a discrete random variable X lists every possible value x together with its probability P(X = x). It answers two questions at once: what values can X take? and how likely is each?

How to build a distribution — four steps

1. Identify the random variable in words. (“Let X = …”)
2. List all possible values of X.
3. Find the probability of each value.
4. Check that all probabilities add up to 1.

Example 1: Flip 2 fair coins, X = number of heads

Sample space: HH, HT, TH, TT (4 equally likely outcomes). X can be 0, 1, or 2.

Example 2: Roll a fair die, X = number rolled

Valid probability distribution rules

• Each probability must satisfy 0 ≤ P(X = x) ≤ 1.
• The probabilities must sum to 1: Σ P(X = x) = 1.

Two rules — both must hold

A discrete probability distribution is valid only when it passes both checks. Miss either one and the table is not a probability distribution.

Valid example

All four values are in [0, 1] and 0.10 + 0.30 + 0.40 + 0.20 = 1.00. Valid.

Invalid example

P(X = 3) = −0.10 is not between 0 and 1, so Rule 1 is violated on the spot.

The poster then adds the four numbers as written — 0.20 + 0.50 + 0.40 + (−0.10) = 1.00 — only to make one narrow point: even if the four cells happen to add to 1, the negative entry still kills the distribution.

What really happens if you ignore the negative entry. Take the three non-negative rows only: 0.20 + 0.50 + 0.40 = 1.10 > 1. So Rule 2 is broken too — the “real” probabilities already sum to more than 1 before the negative row is even considered.

Idea

The probabilities in a distribution must sum to 1. If one probability is missing, use that fact to solve for it: Σ P(X = x) = 1.

Method (three steps)

1. Set up the equation Σ P(X = x) = 1 with the missing probability as an unknown.
2. Add the known probabilities.
3. Solve for the missing probability.

Example 1: Find P(X = 2)

0.20 + 0.30 + P(X = 2) + 0.10 = 1  ⇒  0.60 + P(X = 2) = 1  ⇒  P(X = 2) = 0.40.

Example 2: Find P(Y = 4)

0.15 + 0.25 + 0.35 + P(Y = 4) = 1  ⇒  0.75 + P(Y = 4) = 1  ⇒  P(Y = 4) = 0.25.

What expected value is

The expected value of a random variable is the long-run average outcome if we repeat the random process many times. It is also called the mean of the random variable, μ_X.

Formula

μ_X = E(X) = Σ x · P(X = x)

Multiply each possible value by its probability, then add up the products.

Example 1: Fair six-sided die

E(X) = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 21/6 = 3.5.

You cannot roll a 3.5, but if you roll a die a million times, the average of the results is very close to 3.5.

Example 2: A $2 game

The game costs $2 to play. You win $5 with probability 0.3 and win $0 with probability 0.7. Let X be your net winnings in dollars.

E(X) = (−2)(0.7) + 3(0.3) = −1.4 + 0.9 = −$0.50. In the long run you expect to lose $0.50 per play.

Key takeaways

• Expected value is the long-run average outcome.
• Use E(X) = Σ x · P(X = x).
• It does not have to be a value X can actually take (e.g. 3.5 on a die).
• It can represent an average gain, loss, cost, or number of successes.

What expected value really means

Expected value is the weighted average of all possible outcomes, weighted by their probabilities. It tells us what to expect on average over many repetitions — not what to expect on a single trial.

Example: 3 fair coin flips, X = number of heads

E(X) = 0(1/8) + 1(3/8) + 2(3/8) + 3(1/8) = 12/8 = 1.5.

What does 1.5 heads mean?

You cannot actually get 1.5 heads in one set of 3 flips (you get 0, 1, 2, or 3). But if you repeat the “flip 3 coins” experiment many times, the average number of heads per set will be very close to 1.5.

Three important points

1. Expected value is the long-run average outcome — not the most likely value.
2. It is a weighted average: more likely outcomes pull the answer more.
3. It does not have to be a possible outcome. 1.5 heads, 3.5 on a die, and −$0.50 on a game are all valid expected values you can never actually observe on one trial.

Why expected value matters for decisions

Expected value gives us a single number to compare two random options — an insurance policy vs. going uninsured, playing a game vs. sitting out, choosing route A vs. route B. Whichever choice has the better expected value is, on average, the better long-run decision.

How to interpret the sign

• Positive expected value: favorable — a long-run gain or profit.
• Negative expected value: unfavorable — a long-run loss or extra cost.
• To choose between two options, compute E(X) for each and pick the better one.

Example 1: Carnival game

It costs $2 to play. Win $10 with probability 0.15, or $0 with probability 0.85. Net outcome X:

E(X) = 8(0.15) + (−2)(0.85) = 1.20 − 1.70 = −$0.50 per play. The game is not favorable — over many plays you lose $0.50 each time.

Example 2: Buying shipping insurance

A package has a 5% chance of being lost (which would cost you $80). Insurance costs $3.

• Option A — No insurance: E(cost) = 80(0.05) + 0(0.95) = $4.00.
• Option B — Buy insurance: guaranteed cost = $3.00.

$3 < $4, so buying insurance is the better expected-value decision.

What variance measures

Variance measures how much the possible values of a random variable typically vary around its expected value. It is the expected value of the squared distance from the mean.

Formula

σ_X² = Σ (x − μ_X)² · P(X = x)

Method (three steps)

1. Find the mean μ_X = E(X).
2. For each value, compute (x − μ_X)² · P(X = x).
3. Add all the terms.

Example: Fair six-sided die

μ = 3.5.

Sum: σ_X² = 1.0417 + 0.3750 + 0.0417 + 0.0417 + 0.3750 + 1.0417 = 2.9167.

Comparing two games with the same mean

Game A pays +$10 or −$10 with equal probability. Game B pays +$2 or −$2 with equal probability. Both have E(X) = 0, but Game A has variance 100 and Game B has variance 4. Same expected value, very different risk.

Key facts

• Variance is always nonnegative.
• Larger variance → more spread out, less predictable.
• Smaller variance → more concentrated near the mean.

What standard deviation measures

The standard deviation is the square root of the variance. It measures the typical distance between the values of the random variable and its expected value — in the same units as the variable itself.

Formula

σ_X = √(σ_X²)

Why we need it in addition to variance

Variance is in squared units (dollars squared, points squared) and is hard to interpret directly. Standard deviation puts the answer back in the original units, so it’s the number we actually report.

Example 1: Fair six-sided die

From the previous section, σ_X² = 2.9167. So σ_X = √2.9167 ≈ 1.71. On average, an individual roll differs from the mean 3.5 by about 1.71.

Example 2: Number of heads in 3 fair coin flips

μ_Y = 1.5.

σ_Y² = 0.75, so σ_Y = √0.75 ≈ 0.87 heads.

Key takeaways

• σ_X = √(σ_X²) — same units as the variable.
• Larger σ means more variability; smaller means less.
• Always nonnegative.

What happens when you shift or scale a random variable?

If we add or multiply a random variable by constants, the mean and standard deviation change in a very predictable way. Set Y = a + bX. Then:

Adding a constant (a shift)

Adding a constant slides every outcome up or down by the same amount. The mean shifts by a, and the standard deviation does not change — the spread is unchanged.

Multiplying by a constant (a scale)

Multiplying by b stretches or shrinks every outcome. The mean is multiplied by b, and the standard deviation is multiplied by |b| (we use absolute value because standard deviation is always nonnegative).

Example 1: Adding a constant

Fair die: μ_X = 3.5, σ_X ≈ 1.71. Let Y = X + 4. Then μ_Y = 4 + 3.5 = 7.5, and σ_Y = 1 · 1.71 = 1.71.

Example 2: Multiplying by a constant

Same die. Let Y = 2X. Then μ_Y = 2(3.5) = 7.0, and σ_Y = 2(1.71) = 3.42.

Sums and differences of random variables

We often want to know the mean and standard deviation of a combination of random variables — the total of two dice, the difference of two test scores, the total time of two independent tasks. The rules for the mean are simple; the rules for standard deviation depend on whether the variables are independent.

Mean of a linear combination (always true)

For Z = aX + bY with any constants a and b:

μ_Z = a·μ_X + b·μ_Y

Special cases: μ_X+Y = μ_X + μ_Y and μ_X−Y = μ_X − μ_Y.

Variance rule (only if independent)

If X and Y are independent, the variances add, even for a difference:

σ_Z² = a²σ_X² + b²σ_Y²

So σ_Z = √(a²σ_X² + b²σ_Y²). Note that standard deviations do NOT add — you must square first.

Example 1: Sum of two independent dice

Each die: μ = 3.5, σ ≈ 1.71. For Z = X + Y: μ_Z = 7 and σ_Z = √(1.71² + 1.71²) = √5.8336 ≈ 2.42.

Example 2: Difference of two independent dice

Same dice. For Z = X − Y: μ_Z = 3.5 − 3.5 = 0, but variances still add: σ_Z = √(1.71² + 1.71²) ≈ 2.42.

What if X and Y are NOT independent?

Include the covariance term: σ_Z² = a²σ_X² + b²σ_Y² + 2ab σ_XY. Positive covariance increases the variance of a sum; negative covariance decreases it.