1 1 Slide © 2003 South-Western/Thomson Learning™ Slides Prepared by JOHN S. LOUCKS St. Edward’s University

2 2 Slide © 2003 South-Western/Thomson Learning™ Chapter 6 Continuous Probability Distributions n Uniform Probability Distribution n Normal Probability Distribution n Exponential Probability Distribution  x f(x)f(x)f(x)f(x)

3 3 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ n A random variable is uniformly distributed whenever the probability is proportional to the interval’s length. 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 where: a = smallest value the variable can assume where: a = smallest value the variable can assume b = largest value the variable can assume b = largest value the variable can assume Uniform Probability Distribution

5 5 Slide © 2003 South-Western/Thomson Learning™ Uniform Probability Distribution 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

6 6 Slide © 2003 South-Western/Thomson Learning™ Example: Slater's Buffet n Uniform Probability Distribution 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. The probability density function is f ( x ) = 1/10 for 5 < x < 15 f ( x ) = 1/10 for 5 < x < 15 = 0 elsewhere = 0 elsewhere where: where: x = salad plate filling weight

7 7 Slide © 2003 South-Western/Thomson Learning™ Example: Slater's Buffet n Uniform Probability Distribution 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 /10 Salad Weight (oz.) P(12 < x < 15) = 1/10(3) =.3

8 8 Slide © 2003 South-Western/Thomson Learning™ Example: Slater's Buffet n Expected Value of x E( x ) = ( a + b )/2 = (5 + 15)/2 = (5 + 15)/2 = 10 = 10 n Variance of x Var( x ) = ( b - a ) 2 /12 Var( x ) = ( b - a ) 2 /12 = (15 – 5) 2 /12 = (15 – 5) 2 /12 = 8.33 = 8.33

9 9 Slide © 2003 South-Western/Thomson Learning™ Normal Probability Distribution n Graph of the Normal Probability Density Function  x f(x)f(x)f(x)f(x)

10 Slide © 2003 South-Western/Thomson Learning™ Normal Probability Distribution n Characteristics of the Normal Probability Distribution 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. Two parameters,  (mean) and  (standard deviation), determine the location and shape of the distribution. Two parameters,  (mean) and  (standard deviation), determine the location and shape of the distribution. The highest point on the normal curve is at the mean, which is also the median and mode. The highest point on the normal curve is at the mean, which is also the median and mode. The mean can be any numerical value: negative, zero, or positive. The mean can be any numerical value: negative, zero, or positive. … continued

11 Slide © 2003 South-Western/Thomson Learning™ Normal Probability Distribution n Characteristics of the Normal Probability Distribution The normal curve is symmetric. The normal curve is symmetric. The standard deviation determines the width of the curve: larger values result in wider, flatter curves. The standard deviation determines the width of the curve: larger values result in wider, flatter curves. The total area under the curve is 1 (.5 to the left of the mean and.5 to the right). The total area under the curve is 1 (.5 to the left of the mean and.5 to the right). 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.

12 Slide © 2003 South-Western/Thomson Learning™ Normal Probability Distribution n Percent (%) of Values in Some Commonly Used Intervals 68.26% of values of a normal random variable are within +/- 1 standard deviation of its mean % of values of a normal random variable are within +/- 1 standard deviation of its mean % of values of a normal random variable are within +/- 2 standard deviations of its mean % of values of a normal random variable are within +/- 2 standard deviations of its mean % of values of a normal random variable are within +/- 3 standard deviations of its mean % of values of a normal random variable are within +/- 3 standard deviations of its mean.

13 Slide © 2003 South-Western/Thomson Learning™ Normal Probability Distribution n Normal Probability Density Function where:  = mean  = mean  = standard deviation  = standard deviation  =  = e = e =

14 Slide © 2003 South-Western/Thomson Learning™ 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 .

15 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Compute Standard Normal Probabilities n Excel has two functions for computing probabilities and z values for a standard normal distribution: NORMSDIST is used to compute the cumulative probability given a z value.NORMSDIST is used to compute the cumulative probability given a z value. NORMSINV is used to compute the z value given a cumulative probability.NORMSINV is used to compute the z value given a cumulative probability. (The letter S in the above function names reminds us that they relate to the standard normal probability distribution.)

16 Slide © 2003 South-Western/Thomson Learning™ n Formula Worksheet Using Excel to Compute Standard Normal Probabilities

17 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet Using Excel to Compute Standard Normal Probabilities

18 Slide © 2003 South-Western/Thomson Learning™ n Formula Worksheet Using Excel to Compute Standard Normal Probabilities

19 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet Using Excel to Compute Standard Normal Probabilities

20 Slide © 2003 South-Western/Thomson Learning™ Example: Pep Zone n Standard Normal Probability Distribution 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).

21 Slide © 2003 South-Western/Thomson Learning™ n Standard Normal Probability Distribution The Standard Normal table shows an area of.2967 for the region between the z = 0 and z =.83 lines below. The shaded tail area is = The probability of a stock- out is z = ( x -  )/  = ( )/6 = ( )/6 =.83 =.83 Example: Pep Zone 0.83 Area =.2967 Area =.5 Area = =.2033 =.2033 z

22 Slide © 2003 South-Western/Thomson Learning™ n Using the Standard Normal Probability Table Example: Pep Zone

23 Slide © 2003 South-Western/Thomson Learning™ 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? Let z.05 represent the z value cutting the.05 tail area. Example: Pep Zone Area =.05 Area =.5 Area =.45 0 z.05

24 Slide © 2003 South-Western/Thomson Learning™ 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 = is a reasonable estimate. z.05 = is a reasonable estimate. Example: Pep Zone

25 Slide © 2003 South-Western/Thomson Learning™ n Standard Normal Probability Distribution The corresponding value of x is given by x =  + z.05   = (6) = = A reorder point of 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

26 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Compute Normal Probabilities n Excel has two functions for computing cumulative probabilities and x values for any normal distribution: NORMDIST is used to compute the cumulative probability given an x value. NORMDIST is used to compute the cumulative probability given an x value. NORMINV is used to compute the x value given a cumulative probability. NORMINV is used to compute the x value given a cumulative probability.

27 Slide © 2003 South-Western/Thomson Learning™ n Formula Worksheet for Pep Zone Example Using Excel to Compute Normal Probabilities

28 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet for Pep Zone Example Using Excel to Compute Normal Probabilities Note: P( x > 20) =.2023 here using Excel, while our previous manual approach using the z table yielded.2033 due to our rounding of the z value.

29 Slide © 2003 South-Western/Thomson Learning™ Exponential Probability Distribution n Exponential Probability Density Function for x > 0,  > 0 for x > 0,  > 0 where:  = mean where:  = mean e = e =

30 Slide © 2003 South-Western/Thomson Learning™ Exponential Probability Distribution n Cumulative Exponential Distribution Function where: x 0 = some specific value of x

31 Slide © 2003 South-Western/Thomson Learning™ n Excel’s EXPONDIST function can be used to compute exponential probabilities. n The function has three arguments: First – the value of the random variable x First – the value of the random variable x Second – 1/  (the inverse of the mean number of occurrences in an interval) Second – 1/  (the inverse of the mean number of occurrences in an interval) Third – “TRUE” or “FALSE” (we will always enter “TRUE” because we’re seeking a cumulative probability) Third – “TRUE” or “FALSE” (we will always enter “TRUE” because we’re seeking a cumulative probability) Using Excel to Compute Exponential Probabilities

32 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Compute Exponential Probabilities n Formula Worksheet

33 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet Using Excel to Compute Exponential Probabilities

34 Slide © 2003 South-Western/Thomson Learning™ n Exponential Probability Distribution 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) = /3 = =.4866 Example: Al’s Carwash

35 Slide © 2003 South-Western/Thomson Learning™ Example: Al’s Carwash n Graph of the Probability Density Function x x f(x)f(x) f(x)f(x) P ( x < 2) = area =.4866 Time Between Successive Arrivals (mins.)

36 Slide © 2003 South-Western/Thomson Learning™ n Formula Worksheet for Al’s Carwash Example Using Excel to Compute Exponential Probabilities

37 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet for Al’s Carwash Example Using Excel to Compute Exponential Probabilities

38 Slide © 2003 South-Western/Thomson Learning™ Relationship between the Poisson and Exponential Distributions (If) the Poisson distribution provides an appropriate description of the number of occurrences per interval (If) the Poisson distribution provides an appropriate description of the number of occurrences per interval (If) the exponential distribution provides an appropriate description of the length of the interval between occurrences (If) the exponential distribution provides an appropriate description of the length of the interval between occurrences

39 Slide © 2003 South-Western/Thomson Learning™ End of Chapter 6