Continuous Probability Distributions Many continuous probability distributions including
Continuous Probability Distributions • Many continuous probability distributions, including: ü Uniform ü Normal ü Gamma ü Exponential ü Chi-Squared ü Lognormal ü Weibull PDF MDH Chapter 6 Part 1 v 5 Source: www. itl. nist. gov EGR 252 2015 Slide 1
Uniform Distribution • Simplest – characterized by the interval endpoints, A and B. A≤x≤B =0 elsewhere • Mean and variance: and MDH Chapter 6 Part 1 v 5 EGR 252 2015 Slide 2
Example: Uniform Distribution 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? MDH Chapter 6 Part 1 v 5 EGR 252 2015 Slide 3
Normal Distribution • The “bell-shaped curve” • Also called the Gaussian distribution • The most widely used distribution in statistical analysis ü forms the basis for most of the parametric tests we’ll perform later in this course. ü describes or approximates most phenomena in nature, industry, or research • Random variables (X) following this distribution are called normal random variables. ü the parameters of the normal distribution are μ and σ (sometimes μ and σ2. ) MDH Chapter 6 Part 1 v 5 EGR 252 2015 Slide 4
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) MDH Chapter 6 Part 1 v 5 EGR 252 2015 Slide 5
Standard Normal RV … • Note: the probability of X taking on any value between x 1 and x 2 is given by: • To ease calculations, we define a normal random variable where Z is normally distributed with μ = 0 and σ2 = 1 MDH Chapter 6 Part 1 v 5 EGR 252 2015 Slide 6
Standard Normal Distribution • Table A. 3 Pages 735 -736: “Areas under the Normal Curve” MDH Chapter 6 Part 1 v 5 EGR 252 2015 Slide 7
Examples • P(Z ≤ 1) = • P(Z ≥ -1) = • P(-0. 45 ≤ Z ≤ 0. 36) = MDH Chapter 6 Part 1 v 5 EGR 252 2015 Slide 8
Name: ____________ • 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) = MDH Chapter 6 Part 1 v 5 EGR 252 2015 Slide 9
Applications of the Normal Distribution • A certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard deviation of 2 ohms. What percentage of the resistors will have a resistance less than 44 ohms? • Solution: X is normally distributed with μ = 40 and σ = 2 and x = 44 P(X<44) = P(Z< +2. 0) = 0. 9772 Therefore, we conclude that 97. 72% will have a resistance less than 44 ohms. What percentage will have a resistance greater than 44 ohms? MDH Chapter 6 Part 1 v 5 EGR 252 2015 Slide 10
The Normal Distribution “In Reverse” • Example: Given a normal distribution with μ = 40 and σ = 6, find the value of X for which 45% of the area under the normal curve is to the left of X. Step 1 If P(Z < z) = 0. 45, z = _______ (from Table A. 3) Why is z negative? Step 2 45% X = _____ MDH Chapter 6 Part 1 v 5 EGR 252 2015 Slide 11
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