Continuous Probability Distributions Many continuous probability distributions, including: Uniform Normal Gamma Exponential Chi-Squared Lognormal Weibull Uniform Normal – Gamma Exponential Chi-Squared Lognormal Weibull -

Uniform Distribution Simplest – characterized by the interval endpoints, A and B. A ≤ x ≤ B = 0 elsewhere Mean and variance: and draw distribution

Example A circuit board failure causes a shutdown of a computing system until a new board is delivered. The delivery time X is uniformly distributed between 1 and 5 days. What is the probability that it will take 2 or more days for the circuit board to be delivered? interval = [1,5] f(x) = 1/(B-A) = 1/(5-1) = ¼, 1 < x < 5 (0 elsewhere) First: show the distribution and demonstrate the “intuitive” answer Then – P(X>2) = ∫25 (1/4)dx = 0.75

Normal Distribution The “bell-shaped curve” Also called the Gaussian distribution The most widely used distribution in statistical analysis forms the basis for most of the parametric tests we’ll perform later in this course. describes or approximates most phenomena in nature, industry, or research Random variables (X) following this distribution are called normal random variables. the parameters of the normal distribution are μ and σ (sometimes μ and σ2.) note: nonparametric tests are distribution-free, assume no underlying distribution (see ch 16)

Normal Distribution The density function of the normal random variable X, with mean μ and variance σ2, is all x. (μ = 5, σ = 1.5) properties of the curve: peak is both the mean and the mode and occurs at x = μ curve is symmetrical about a vertical axis through the mean total area under the curve and above the horizontal axis = 1. points of inflection are at x = μ + σ

Standard Normal RV … Note: the probability of X taking on any value between x1 and x2 is given by: To ease calculations, we define a normal random variable where Z is normally distributed with μ = 0 and σ2 = 1

Standard Normal Distribution Table A.3: “Areas Under the Normal Curve”

Examples P(Z ≤ 1) = P(Z ≥ -1) = P(-0.45 ≤ Z ≤ 0.36) =

Your turn … Use Table A.3 to determine (draw the picture!) 1. P(Z ≤ 0.8) = 2. P(Z ≥ 1.96) = 3. P(-0.25 ≤ Z ≤ 0.15) = 4. P(Z ≤ -2.0 or Z ≥ 2.0) =