Probability Distributions

A random variable X is

A random variable X is a function that assigns numerical values to outcomes of a random experiment. It bridges the gap between the abstract sample spa

A random variable that takes on

A random variable that takes on a countable number of values (finite or infinite). Examples include the number of heads in 5 coin tosses (0, 1, 2, 3,

For a discrete random variable X

For a discrete random variable X taking values x₁, x₂, x₃, ..., the PMF is f(x) = P(X = x). It must satisfy: (1) f(x) ≥ 0 for all x, and (2) Σf(x) = 1

For any random variable

For any random variable, F(x) = P(X ≤ x) is the CDF. For discrete variables: F(x) = Σ[xᵢ ≤ x] f(xᵢ). Key properties: (1) F is non-decreasing, (2) 0 ≤

A random variable that can take

A random variable that can take any value in an interval (like weight, height, lifetime of equipment). Key property: P(X = x) = 0 for any specific val