Random Variables and Probability Distributions

A random variable X is

A random variable X is a function that assigns real numbers to outcomes of a random experiment. For example, when tossing two coins, if X represents t

Discrete random variables take finite

Discrete random variables take finite or countably infinite values (like number of heads in coin tosses: 0, 1, 2, ...). Continuous random variables ca

A probability distribution shows all possible

A probability distribution shows all possible values of a random variable and their corresponding probabilities. Must satisfy: (1) P(xi) ≥ 0 for all v

Mean μ = Σ xi P(xi)

Mean μ = Σ xi P(xi) represents the average value expected from the random variable. Step-by-step calculation: (1) Multiply each value by its probabili

Variance σ² = Σ (xi

Variance σ² = Σ (xi - μ)² P(xi) = Σ xi² P(xi) - μ² measures spread around the mean. Standard deviation σ = √variance. Two-step calculation method: Met