Understanding Discrete Probability Distributions: Random Variables, Binomial Experiments, , Study notes of Statistics

Download Understanding Discrete Probability Distributions: Random Variables, Binomial Experiments, and more Study notes Statistics in PDF only on Docsity! STA 100 Lecture 5 Discrete Probability Distributions Random Variables A random variable is a variable whose numerical values are determined by the outcomes of a random experiment. Examples: Types of random variables: 1. Discrete. 2. Continuous. Probability Distribution of a Discrete Random Variable The probability distribution of a discrete random variable Y is represented by a probability function p(y) defined as: p(y) = P[Y = y] Example: Cost of health care Charges (y) p(y) Clinic $ 50 0.60 Clinic and Lab $ 80 0.30 ER visit $ 160 0.10 Properties of probability distribution. Expected Value and Variance Let Y be a discrete random variable. The expected value (mean) of Y is defined as: μ = E(Y) = Σ y p(y) Example: Cost of health care If Y is a random variable, then E(a Y + b) = a E(Y) + b. Example: Cost of health care. Variance of a random variable Y is defined as σ2 = E(Y - μ)2 = Σ (y - μ)2 p(y) Example: Cost of health care If Y is a random variable, then Var(a Y + b) = a2 Var(Y) Example: Cost of health care If X and Y are two independent random variables, then Var( X + Y ) = Var (X) + Var (Y) Var( aX + bY + c) = a2 Var(X) + b2 Var(Y) 2