Continuous Probability Distributions Part 2 Many continuous probability

Continuous Probability Distributions Part 2 • Many continuous probability distributions, including: ü Uniform ü Normal ü Gamma ü Exponential ü Chi-Squared ü Lognormal ü Weibull JMB Chapter 6 Part 2 EGR 252 Spring 2013 Slide 1

Review: Standard Normal Random Variable • Normal Distribution Review: 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 JMB Chapter 6 Part 2 EGR 252 Spring 2013 Slide 2

Review: Standard Normal Distribution • Table A. 3 Pages 735 -736: “Areas under the Normal Curve” JMB Chapter 6 Part 2 EGR 252 Spring 2013 Slide 3

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? JMB Chapter 6 Part 2 EGR 252 Spring 2013 Slide 4

Gamma & Exponential Distributions • Related to the Poisson Process (discrete Ch. 5) üNumber of occurrences in a given interval or region ü“Memoryless” process • Sometimes we’re interested in the number of events that occur in an area (eg flaws in a square yard of cotton). • Sometimes we’re interested in the time until a certain number of events occur. • Area and time are variables that are measured (continuous). JMB Chapter 6 Part 2 EGR 252 Spring 2013 Slide 5

Gamma Distribution • The density function of the random variable X with gamma distribution having parameters α (number of occurrences) and β (time or region). x > 0. μ = αβ σ2 = αβ 2 JMB Chapter 6 Part 2 EGR 252 Spring 2013 Slide 6

Exponential Distribution • Special case of the gamma distribution with α = 1. x > 0. ü Describes the time until Poisson event ü Describes the time between Poisson events μ=β σ2 = β 2 JMB Chapter 6 Part 2 EGR 252 Spring 2013 Slide 7

Is It a Poisson Process? • For homework and exams in the introductory statistics course, you will be told that the process is Poisson. ü An average of 2. 7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process. What is the probability that up to a minute will elapse before 2 calls arrive? ü An average of 2. 7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process. How long before the next call? JMB Chapter 6 Part 2 EGR 252 Spring 2013 Slide 8

Poisson Example Problem An average of 2. 7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process. What is the probability that up to 1 minute will elapse before 2 calls arrive? ü β = 1 / λ = 1 / 2. 7 = 0. 3704 ü α=2 ü x=1 What is the value of P(X ≤ 1)? Can we use a table? No We must use integration. JMB Chapter 6 Part 2 EGR 252 Spring 2013 Slide 9

Poisson Example Solution An average of 2. 7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process. What is the probability that up to 1 minute will elapse before 2 calls arrive? The time until a number of Poisson events occurs follows the gamma distribution. β = 1/ 2. 7 = 0. 3704 α = 2 (calls) P(X < 1) = (1/ β 2) x e-x/ β dx = 2. 72 x e -2. 7 x dx = [-2. 7 xe-2. 7 x – e-2. 7 x] 01 = 1 – e-2. 7 (1 + 2. 7) = 0. 7513 Using Excel: =GAMMADIST(1, 2, 1/2. 7, TRUE) JMB Chapter 6 Part 2 EGR 252 Spring 2013 Slide 10

Another Type of Question An average of 2. 7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process. What is the expected time before the next call arrives? Expected value = μ = α β α = 1(call) β = 1/2. 7 μ = β = 0. 3704 min. We expect the next call to arrive in 0. 3704 minutes. When α = 1 the gamma distribution is known as the exponential distribution. JMB Chapter 6 Part 2 EGR 252 Spring 2013 Slide 11