JMB Chapter 6 Lecture 3 EGR 252 Spring 2011 Slide 1 Continuous Probability Distributions Many continuous probability distributions, including: Uniform Normal Gamma Exponential Chi-Squared Lognormal Weibull

JMB Chapter 6 Lecture 3 EGR 252 Spring 2011 Slide 2 Standard Normal Distribution Table A.3: “Areas Under the Normal Curve”

JMB Chapter 6 Lecture 3 EGR 252 Spring 2011 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) = 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 Lecture 3 EGR 252 Spring 2011 Slide 4 The Normal Distribution “In Reverse” Example: Given a normal distribution with = 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. Solution: If P(Z < k) = 0.45, what is the value of k? 45% of area under curve less than k k = from table in back of book Z = (x- μ ) / σ Z = = (x-40) / 6 X = X values Z values

JMB Chapter 6 Lecture 3 EGR 252 Spring 2011 Slide 5 Normal Approximation to the Binomial If n is large and p is not close to 0 or 1, or if n is smaller but p is close to 0.5, then the binomial distribution can be approximated by the normal distribution using the transformation: NOTE: When we apply the theorem, we apply a continuity correction: add or subtract 0.5 from X to be sure the value of interest is included (drawing a picture helps you determine whether to add or subtract To find the area under the normal curve to the left of x+.0.5, use:

JMB Chapter 6 Lecture 3 EGR 252 Spring 2011 Slide 6 Look at example 6.15, pg The probability that a patient recovers is p = 0.4 If 100 people have the disease, what is the probability that less then 30 survive? n = 100 μ = np =____σ = sqrt(npq)= ____ if x = 30, then z = _____________ (Don’t forget the continuity correction.) and, P(X < 30) = P (Z < _________) = _________ (Less than 30 for a discrete distribution is 29 or less. Adding.5 to 29 insures we “cover” the maximum value.)

JMB Chapter 6 Lecture 3 EGR 252 Spring 2011 Slide 7 The Normal Approximation: Your Turn Using μ and σ from the previous example, 1. What is the probability that more than 50 survive? More than 50 for a discrete distribution is 51 or greater. How do we include 51 and not 50 ? DRAW THE PICTURE!!

JMB Chapter 6 Lecture 3 EGR 252 Spring 2011 Slide 8 The Normal Approximation: Your Turn 2 Using μ and σ from the previous example, 1. What is the probability that exactly 45 survive if we use the normal approximation to the binomial? Hint: Find the area under the curve between two values. 2. What is the probability that exactly 45 survive if we use the binomial distribution? b(45;100,0.4) DRAW THE PICTURE!!

JMB Chapter 6 Lecture 3 EGR 252 Spring 2011 Slide 9 Gamma & Exponential Distributions Related to the Poisson Process (discrete) Number of occurrences in a given interval or region “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. Typical problem statement: The length of time in days between student complaints about too much homework follows a gamma distribution with α = 3 and β = 7. What is the probability that up to two weeks will elapse before 6 complaints are heard?

JMB Chapter 6 Lecture 3 EGR 252 Spring 2011 Slide 10 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 σ 2 = αβ 2

JMB Chapter 6 Lecture 3 EGR 252 Spring 2011 Slide 11 Exponential Distribution Special case of the gamma distribution with α = 1. x > 0. Describes the time until or time between Poisson events. μ = β σ 2 = β 2

JMB Chapter 6 Lecture 3 EGR 252 Spring 2011 Slide 12 Is It a Poisson Process? For homework and exams in the introductory statistics course, you will be told that the process is Poisson. 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? An average of 3.5 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 Lecture 3 EGR 252 Spring 2011 Slide 13 Poisson Example Problem 1 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? β = 1 / λ = 1 / 2.7 = α = 2 X=1 What is the value of P(X ≤ 1)? Can we use a table? Must we integrate? Can we use Excel?

JMB Chapter 6 Lecture 3 EGR 252 Spring 2011 Slide 14 Poisson Example Solution Based on the Gamma Distribution 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? β = 1/ 2.7 = α = 2 = Excel = GAMMADIST (1, 2, , TRUE)

JMB Chapter 6 Lecture 3 EGR 252 Spring 2011 Slide 15 Poisson Example Problem 2 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? α = next call =1 λ = 1/ β = 2.7 Expected value of time (Gamma) = μ = α β Since α =1 μ = β = 1 / 2.7 = min. Note that when α = 1 the gamma distribution is known as the exponential distribution.