AP Statistics Unit 4: Random Variables

Random Variable

A numerical description of the outcome of a statistical experiment.

Discrete Random Variable

A variable that has a countable number of possible values.

Continuous Random Variable

A variable that can take any value within an interval on the number line.

Probability Distribution

Lists all possible values of a discrete random variable and their corresponding probabilities.

Valid Probability Distribution Requirements

Every probability p_i must be between 0 and 1, and the sum of all probabilities must equal 1.

Expected Value

The long-run average outcome of a random variable, denoted as E(X).

Mean of a Discrete Random Variable formula

μX = E(X) = ∑ xi pi, where xi are possible values and p_i are probabilities.

Variance

A measure of the variability or spread of a random variable.

Variance Formula

Var(X) = σX² = ∑ (xi - μX)² pi.

Standard Deviation Formula

σX = √(∑ (xi - μX)² pi).

Linear Transformation

An equation applied to a random variable in the form Y = a + bX.

Effect on Mean when adding constant a

Changes the mean.

Effect on Variance when multiplying by b

Changes by b².

Expected Value of a Sum of Independent Random Variables

μ{X+Y} = μX + μ_Y.

Variance of a Sum of Independent Random Variables

σ²{X±Y} = σ²X + σ²_Y.

Key to Combining Variances

Independence of random variables is required for this rule.

Pythagorean Theorem of Statistics for Standard Deviation

σ{X±Y} = √(σ²X + σ²_Y).

Common Mistake: Adding Standard Deviations

Mistakenly calculating σ{X+Y} = σX + σ_Y.

Correction for Adding Standard Deviations

Always square first, add variances, then take the square root.

Common Mistake: Subtracting Variances

Calculating σ²D = σ²X - σ²_Y for difference D = X - Y.

Key Correction for Subtracting Variances

Always add variances: σ²D = σ²X + σ²_Y.

Continuous Probability

In continuous distributions, probability at a specific point is always 0.

Example of Discrete Random Variable

Number of heads in 3 coin flips.

Mean of a Raffle Ticket Example

E(X) = -$9.00, the expected loss per ticket.

Understanding Independence in Random Variables

Assume independence when calculating combined variances.