1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 6 Continuous Probability Distributions n Uniform Probability Distribution n Normal Probability Distribution n Exponential Probability Distribution f ( x ) x x Uniform x Normal x x Exponential

3 3 Slide © 2008 Thomson South-Western. All Rights Reserved n Converting to the Standard Normal Distribution Standard Normal Probability Distribution We can think of z as a measure of the number of standard deviations x is from .

4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Standard Normal Probability Distribution n Example: Pep Zone Pep Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is placed. Pep Zone 5w-20 Motor Oil

5 5 Slide © 2008 Thomson South-Western. All Rights Reserved The store manager is concerned that sales are being The store manager is concerned that sales are being lost due to stockouts while waiting for an order. It has been determined that demand during replenishment lead-time is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons. The manager would like to know the probability of a stockout, P ( x > 20). Standard Normal Probability Distribution Pep Zone 5w-20 Motor Oil n Example: Pep Zone

6 6 Slide © 2008 Thomson South-Western. All Rights Reserved z = ( x -  )/  z = ( x -  )/  = ( )/6 = ( )/6 =.83 =.83 z = ( x -  )/  z = ( x -  )/  = ( )/6 = ( )/6 =.83 =.83 n Solving for the Stockout Probability Step 1: Convert x to the standard normal distribution. Pep Zone 5w-20 Motor Oil Step 2: Find the area under the standard normal curve to the left of z =.83. curve to the left of z =.83. Step 2: Find the area under the standard normal curve to the left of z =.83. curve to the left of z =.83. see next slide see next slide Standard Normal Probability Distribution

7 7 Slide © 2008 Thomson South-Western. All Rights Reserved n Cumulative Probability Table for the Standard Normal Distribution Pep Zone 5w-20 Motor Oil P ( z <.83) Standard Normal Probability Distribution

8 8 Slide © 2008 Thomson South-Western. All Rights Reserved P ( z >.83) = 1 – P ( z.83) = 1 – P ( z <.83) = = =.2033 =.2033 P ( z >.83) = 1 – P ( z.83) = 1 – P ( z <.83) = = =.2033 =.2033 n Solving for the Stockout Probability Step 3: Compute the area under the standard normal curve to the right of z =.83. curve to the right of z =.83. Step 3: Compute the area under the standard normal curve to the right of z =.83. curve to the right of z =.83. Pep Zone 5w-20 Motor Oil Probability of a stockout of a stockout P ( x > 20) Standard Normal Probability Distribution

9 9 Slide © 2008 Thomson South-Western. All Rights Reserved n Solving for the Stockout Probability 0.83 Area =.7967 Area = =.2033 =.2033 z Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution

10 Slide © 2008 Thomson South-Western. All Rights Reserved n Standard Normal Probability Distribution If the manager of Pep Zone wants the probability of a stockout to be no more than.05, what should the reorder point be? Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution

11 Slide © 2008 Thomson South-Western. All Rights Reserved n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil 0 Area =.9500 Area =.0500 z z.05 Standard Normal Probability Distribution

12 Slide © 2008 Thomson South-Western. All Rights Reserved n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil Step 1: Find the z -value that cuts off an area of.05 in the right tail of the standard normal in the right tail of the standard normal distribution. distribution. Step 1: Find the z -value that cuts off an area of.05 in the right tail of the standard normal in the right tail of the standard normal distribution. distribution. We look up the complement of the tail area ( =.95) Standard Normal Probability Distribution

13 Slide © 2008 Thomson South-Western. All Rights Reserved n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil Step 2: Convert z.05 to the corresponding value of x. x =  + z.05  x =  + z.05   = (6) = or 25 = or 25 x =  + z.05  x =  + z.05   = (6) = or 25 = or 25 A reorder point of 25 gallons will place the probability A reorder point of 25 gallons will place the probability of a stockout during leadtime at (slightly less than).05. of a stockout during leadtime at (slightly less than).05. Standard Normal Probability Distribution

14 Slide © 2008 Thomson South-Western. All Rights Reserved n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil By raising the reorder point from 20 gallons to By raising the reorder point from 20 gallons to 25 gallons on hand, the probability of a stockout decreases from about.20 to.05. This is a significant decrease in the chance that Pep This is a significant decrease in the chance that Pep Zone will be out of stock and unable to meet a customer’s desire to make a purchase. Standard Normal Probability Distribution

15 Slide © 2008 Thomson South-Western. All Rights Reserved Normal Approximation of Binomial Probabilities When the number of trials, n, becomes large, When the number of trials, n, becomes large, evaluating the binomial probability function by hand or with a calculator is difficult The normal probability distribution provides an The normal probability distribution provides an easy-to-use approximation of binomial probabilities where n > 20, np > 5, and n (1 - p ) > 5.

16 Slide © 2008 Thomson South-Western. All Rights Reserved Normal Approximation of Binomial Probabilities Set  = np Set  = np n Add and subtract 0.5 (a continuity correction factor) because a continuous distribution is being used to because a continuous distribution is being used to approximate a discrete distribution. For example, approximate a discrete distribution. For example, P ( x = 10) is approximated by P (9.5 < x < 10.5). P ( x = 10) is approximated by P (9.5 < x < 10.5).

17 Slide © 2008 Thomson South-Western. All Rights Reserved Exponential Probability Distribution n The exponential probability distribution is useful in describing the time it takes to complete a task. n The exponential random variables can be used to describe: Time between vehicle arrivals at a toll booth Time between vehicle arrivals at a toll booth Time required to complete a questionnaire Time required to complete a questionnaire Distance between major defects in a highway Distance between major defects in a highway SLOW

18 Slide © 2008 Thomson South-Western. All Rights Reserved n Density Function Exponential Probability Distribution where:  = mean e = e = for x > 0,  > 0

19 Slide © 2008 Thomson South-Western. All Rights Reserved n Cumulative Probabilities Exponential Probability Distribution where: x 0 = some specific value of x x 0 = some specific value of x

20 Slide © 2008 Thomson South-Western. All Rights Reserved Exponential Probability Distribution n Example: Al’s Full-Service Pump The time between arrivals of cars The time between arrivals of cars at Al’s full-service gas pump follows an exponential probability distribution with a mean time between arrivals of 3 minutes. Al would like to know the probability that the time between two successive arrivals will be 2 minutes or less.

21 Slide © 2008 Thomson South-Western. All Rights Reserved x x f(x)f(x) f(x)f(x) Time Between Successive Arrivals (mins.) Exponential Probability Distribution P ( x < 2) = /3 = =.4866 P ( x < 2) = /3 = =.4866

22 Slide © 2008 Thomson South-Western. All Rights Reserved Relationship between the Poisson and Exponential Distributions The Poisson distribution provides an appropriate description of the number of occurrences per interval The Poisson distribution provides an appropriate description of the number of occurrences per interval The exponential distribution provides an appropriate description of the length of the interval between occurrences The exponential distribution provides an appropriate description of the length of the interval between occurrences

23 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 6