2.7 Introduction to Random Variables and Probability Distributions

What is a random variable?

A random variable assigns a numeric value to each outcome of a chance process. A discrete random variable takes countable values (e.g., number of heads in 3 flips), while a continuous random variable takes any value in an interval (e.g., height). Each is paired with a probability distribution describing how likely each value or range of values is.

Probability distributions

A discrete probability distribution lists each possible value with its probability; probabilities must be between 0 and 1 and sum to 1. Distributions can be shown in tables, histograms, or formulas. For continuous random variables, probabilities are found using areas under a density curve rather than individual point probabilities.

Mean and interpreting distributions

The mean (expected value) of a discrete random variable, μₓ = Σ[x·P(x)], is the long-run average outcome over many repetitions — a weighted average using probabilities as weights. It doesn't need to equal a possible outcome. Distribution shape, center, and spread help describe overall behavior.

Key terms

Random variable

A variable that assigns a numeric value to each outcome of a chance process.

Discrete random variable

A random variable with countable possible values (e.g., number of successes).

Continuous random variable

A random variable that can take any value within an interval, with probabilities given by areas under a density curve.

Valid probability distribution

All probabilities are between 0 and 1, and the total sums to 1.

Expected value (mean) of a random variable

μₓ = Σ[x·P(x)], the long-run average outcome over many repetitions.

Try an AP-style question

What is P(X = 3)?

1. 0.4
2. 0.1
3. 0.6
4. 0.3