1. 1 1 Slide © 2001 South-Western College Publishing/Thomson Learning Anderson Sweeney Williams Anderson Sweeney Williams Slides Prepared by JOHN LOUCKS QUANTITATIVE METHODS FOR BUSINESS 8e QUANTITATIVE METHODS FOR BUSINESS 8e

2. 2 2 Slide Chapter 3, Part II Continuous Random Variables n Continuous Random Variables n Normal Probability Distribution n Exponential Probability Distribution

3. 3 3 Slide Continuous Probability Distributions n A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. n It is not possible to talk about the probability of the random variable assuming a particular value. n Instead, we talk about the probability of the random variable assuming a value within a given interval. n The probability of the random variable assuming a value within some given interval from x 1 to x 2 is defined to be the area under the graph of the probability density function between x 1 and x 2.

4. 4 4 Slide Uniform Probability Distribution A random variable is uniformly distributed whenever the probability is proportional to the length of the interval. n Uniform Probability Density Function f ( x ) = 1/( b - a ) for a < x < b f ( x ) = 1/( b - a ) for a < x < b = 0 elsewhere = 0 elsewhere n Expected Value of x E( x ) = ( a + b )/2 n Variance of x Var( x ) = ( b - a ) 2 /12 Var( x ) = ( b - a ) 2 /12 where: a = smallest value the variable can assume b = largest value the variable can assume b = largest value the variable can assume

5. 5 5 Slide Example: Slater's Buffet Slater customers are charged for the amount of salad they take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces. n Probability Density Function f ( x ) = 1/10 for 5 < x < 15 = 0 elsewhere = 0 elsewherewhere x = salad plate filling weight

6. 6 6 Slide Example: Slater's Buffet What is the probability that a customer will take between 12 and 15 ounces of salad? f(x)f(x) f(x)f(x) x x 5 5 10 15 12 1/10 Salad Weight (oz.) P(12 < x < 15) = 1/10(3) =.3

7. 7 7 Slide The Normal Probability Distribution n Graph of the Normal Probability Density Function  x f(x)f(x)f(x)f(x)

8. 8 8 Slide Normal Probability Distribution n The Normal Curve The shape of the normal curve is often illustrated as a bell-shaped curve.The shape of the normal curve is often illustrated as a bell-shaped curve. The highest point on the normal curve is at the mean, which is also the median and mode of the distribution.The highest point on the normal curve is at the mean, which is also the median and mode of the distribution. The normal curve is symmetric.The normal curve is symmetric. The standard deviation determines the width of the curve.The standard deviation determines the width of the curve. The total area under the curve is 1.The total area under the curve is 1. Probabilities for the normal random variable are given by areas under the curve.Probabilities for the normal random variable are given by areas under the curve.

9. 9 9 Slide Normal Probability Distribution n Normal Probability Density Function where  = mean  = mean  = standard deviation  = standard deviation  = 3.14159  = 3.14159 e = 2.71828 e = 2.71828

10. 10 Slide Standard Normal Probability Distribution n A random variable that has a normal distribution with a mean of zero and a standard deviation of one is said to have a standard normal probability distribution. n The letter z is commonly used to designate this normal random variable. n Converting to the Standard Normal Distribution We can think of z as a measure of the number of standard deviations x is from . We can think of z as a measure of the number of standard deviations x is from .

11. 11 Slide 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. The store manager is concerned that sales are being lost due to stockouts while waiting for an order. It has been determined that leadtime demand 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).

12. 12 Slide n Standard Normal Distribution Distribution z = ( x -  )/  = (20 - 15)/6 = (20 - 15)/6 =.83 =.83 The Standard Normal table shows an area of.2967 for the region between the z = 0 line and the z =.83 line above. The shaded tail area is.5 -.2967 =.2033. The probability of a stockout is.2033. 0.83 Area =.2967 Area =.5 Area =.2033 z Example: Pep Zone

13. 13 Slide n Using the Standard Normal Probability Table Example: Pep Zone

14. 14 Slide Example: Pep Zone n Using an Excel Spreadsheet Step 1: Select a cell in the worksheet where you want the normal probability to appear.Step 1: Select a cell in the worksheet where you want the normal probability to appear. Step 2: Select the Insert pull-down menu.Step 2: Select the Insert pull-down menu. Step 3: Choose the Function option.Step 3: Choose the Function option. Step 4: When the Paste Function dialog box appears: Choose Statistical from the Function Category box. Choose NORMDIST from the Function Name box. Select OK.Step 4: When the Paste Function dialog box appears: Choose Statistical from the Function Category box. Choose NORMDIST from the Function Name box. Select OK. continue

15. 15 Slide Example: Pep Zone n Using an Excel Spreadsheet (continued) Step 5: When the NORMDIST dialog box appears:Step 5: When the NORMDIST dialog box appears: Enter 20 in the x box. Enter 15 in the mean box. Enter 15 in the mean box. Enter 6 in the standard deviation box. Enter true in the cumulative box. Select OK. At this point,.7967 will appear in the cell selected in Step 1, indicating that the probability of demand during lead time being less than or equal to 20 gallons is.7967. The probability that demand will exceed 20 gallons is 1 -.7967 =.2033.

16. 16 Slide If the manager of Pep Zone wants the probability of a stockout to be no more than.05, what should the reorder point be? Let z.05 represent the z value cutting the tail area of.05. Area =.05 Area =.5 Area =.45 0 z.05 Example: Pep Zone

17. 17 Slide n Using the Standard Normal Probability Table We now look-up the.4500 area in the Standard Normal Probability table to find the corresponding z.05 value. z.05 = 1.645 is a reasonable estimate. z.05 = 1.645 is a reasonable estimate. Example: Pep Zone

18. 18 Slide The corresponding value of x is given by x =  + z.05   = 15 + 1.645(6) = 24.87 = 24.87 A reorder point of 24.87 gallons will place the probability of a stockout during leadtime at.05. Perhaps Pep Zone should set the reorder point at 25 gallons to keep the probability under.05. Example: Pep Zone

19. 19 Slide Exponential Probability Distribution n Exponential Probability Density Function for x > 0,  > 0 for x > 0,  > 0 where  = mean e = 2.71828 e = 2.71828 n Cumulative Exponential Distribution Function where x 0 = some specific value of x

20. 20 Slide The time between arrivals of cars at Al’s Carwash 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. P( x < 2) = 1 - 2.71828 -2/3 = 1 -.5134 =.4866 Example: Al’s Carwash

21. 21 Slide Example: Al’s Carwash n Graph of the Probability Density Function x x f(x)f(x) f(x)f(x).1.3.4.2 1 2 3 4 5 6 7 8 9 10 P( x < 2) = area =.4866

22. 22 Slide Relationship Between the Poisson and Exponential Distributions n The continuous exponential probability distribution is related to the discrete Poisson distribution. n The Poisson distribution provides an appropriate description of the number of occurrences per interval. n The exponential distribution provides a description of the length of the interval between occurrences.

23. 23 Slide The End of Chapter 3, Part II