Random Variables & Distributions: every key term you need (+ practice quiz)

More AP Statistics guides

• Designing Studies
• Exploring Bivariate Data
• Probability
• Sampling Distributions
• Inference for Means

Random Variables

Function assigning numerical value to each outcome in sample space. Discrete (countable values) or continuous (any value in range).

Probability Distribution (Discrete)

Table/function listing all possible values and their probabilities. Must sum to 1. Described by mean μ and standard deviation σ.

Expected Value μ

μ = E(X) = Σ x·P(x). Long-run average; center of distribution. Linear transformation: E(aX+b) = aE(X) + b.

Variance σ²

σ² = E[(X-μ)²] = Σ(x-μ)²·P(x). Measures spread. σ² = E(X²) - [E(X)]².

Standard Deviation σ

σ = √(σ²). Same units as data. Measures typical deviation from mean.

Binomial Distribution

n independent trials, each success probability p. X = number of successes. Conditions: fixed n, independent trials, constant p, binary outcomes.

Binomial Probability

P(X=k) = C(n,k)·p^k·(1-p)^(n-k). Binomial: μ = np, σ² = np(1-p), σ = √(np(1-p)).

Geometric Distribution

Number of trials until first success; each trial has probability p of success. P(X=k) = (1-p)^(k-1)·p.

Geometric Mean & Variance

μ = 1/p, σ² = (1-p)/p². Approaches ∞ as p→0 (rare success = many trials needed).

Poisson Distribution

Number of events in fixed time/space interval when events occur at constant rate λ. P(X=k) = (e^-λ·λ^k) / k!.

Normal Distribution

Bell-shaped, symmetric, continuous. 68% within 1σ, 95% within 2σ, 99.7% within 3σ of μ. Standard normal: μ=0, σ=1.

Z-Score Standardization

z = (x-μ)/σ. Converts any normal to standard normal. Enables comparison across distributions.

Normal Probability Calculations

Use z-table or calculator to find P(X≤x), P(a≤X≤b), etc. Convert to standard normal first.

Inverse Normal

Find x-value given probability. Example: 90th percentile. Use inverse function on calculator.

Transformation of Random Variables

If Y = aX+b, then E(Y) = aE(X)+b, Var(Y) = a²Var(X). Linear transforms scale mean and variance predictably.

Sum of Independent Variables

E(X+Y) = E(X)+E(Y). If independent: Var(X+Y) = Var(X)+Var(Y). Central Limit Theorem makes sum approach normal.

Unit 4 Summary

Discrete distributions: binomial, geometric, Poisson. Continuous: normal. Expected value, variance, transformations predict distribution behavior. Normal distribution dominates statistics.

Expected Value of Discrete RV

μ_X = Σ x_i · p_i. The long-run average value over many repetitions; need not be a possible value of X.

Variance of Discrete RV

σ_X^2 = Σ (x_i - μ_X)^2 · p_i; σ_X is its square root. Measures typical distance of outcomes from the mean.

Linear Transformation Y = a + bX

μ_Y = a + b·μ_X, σ_Y = |b|·σ_X. Adding a constant shifts center only; multiplying scales both center and spread.

Sum/Difference of Independent RVs

μ_{X±Y} = μ_X ± μ_Y always. Variances ADD for both sums and differences when independent: σ^2_{X±Y} = σ_X^2 + σ_Y^2.

Never Add SDs

Standard deviations do not add; add variances then take the square root. σ_{X+Y} = sqrt(σ_X^2 + σ_Y^2).

Binomial Setting (BINS)

Binary outcomes, Independent trials, fixed Number of trials n, Same probability p on each trial. X = number of successes.

Binomial Formula

P(X = k) = C(n,k) p^k (1-p)^(n-k). μ = np, σ = sqrt(np(1-p)).

10% Condition

When sampling without replacement, trials are approximately independent if the sample is less than 10% of the population, so binomial applies.

Large Counts Condition

A binomial distribution is approximately Normal when np ≥ 10 and n(1-p) ≥ 10.

Geometric Setting

Same as binomial but no fixed n; X = number of trials until the FIRST success. P(X = k) = (1-p)^(k-1) p, μ = 1/p.

Geometric Shape

Always right-skewed with the highest probability at X = 1, since each additional failure multiplies the probability by (1-p).

Binomial Cumulative Probability

P(X ≤ k) via binomcdf(n, p, k). P(X ≥ k) = 1 - binomcdf(n, p, k-1). P(X > k) = 1 - binomcdf(n, p, k).

Continuous Random Variable

Takes all values in an interval; probability is area under a density curve. P(X = any single value) = 0, so ≤ and < are equivalent.

Normal Probability

Standardize z = (x - μ)/σ, then use Table A or normalcdf(lower, upper, μ, σ). Always sketch and shade.

Inverse Normal

Given a percentile, find x: invNorm(area, μ, σ) or find z from the table then x = μ + z·σ.

Sum of Independent Normals

If X and Y are independent Normal, X + Y and X - Y are Normal with mean μ_X ± μ_Y and SD sqrt(σ_X^2 + σ_Y^2).

Combining Non-Independent RVs

Means still add, but the variance rule requires independence. Without it, σ_{X+Y}^2 also involves the covariance and cannot be found from σ_X, σ_Y alone.

Expected Value in Games

Fair game: expected net winnings = 0. Casino games have negative expected value for the player; small per-play losses accumulate by LLN.

Binomial vs Geometric Question Cue

'How many successes in n trials?' is binomial. 'How many trials until the first success?' is geometric.

Mean vs Expected Value Interpretation

'If we repeated the random process many times, the average value of X would be about μ_X.' Always in context.

Discrete Probability Histogram

Bar heights are probabilities; total height sums to 1. Use to describe shape (skew, symmetry) of a discrete distribution.