Probability · Quick Reference

Random Variables Cheat Sheet

Probability · Lesson 5/8

In one line: a random variable assigns a number to every outcome of a random experiment, turning abstract outcomes into something we can calculate with.

Key Ideas

1Random Variable (X). A rule that assigns a numeric value to each outcome in a sample space, such as X = number of heads in 3 coin flips.

2Discrete Random Variables. Take on a countable set of values, often whole numbers (number of defective items, number of customers).

3Continuous Random Variables. Can take any value within a range, often from measurements (height, weight, time).

4Probability Distribution. A table, formula, or graph showing every possible value of X and its associated probability.

5Expected Value (Mean). E(X) = sum of [x * P(x)] over all possible values — the long-run average value of X.

Worked Examples

A coin is flipped twice. Let X = number of heads. Find the probability distribution of X.

P(X=0)=1/4, P(X=1)=1/2, P(X=2)=1/4

Using the distribution above, find E(X).

E(X) = 1

A random variable X takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3. Find E(X).

E(X) = 2.1

Formulas

E(X) = sum of [x * P(x)]

Var(X) = E(X^2) - [E(X)]^2

Practice Yourself

Is 'number of cars passing an intersection in an hour' discrete or continuous?

Discrete

Is 'exact weight of a bag of rice' discrete or continuous?

Continuous

X takes values 0,1 with P(0)=0.3, P(1)=0.7. Find E(X).

0.7