Random Variables & Distributions: every key term you need

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.

More AP Statistics guides

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