# 1 1 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole.

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1 1 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Slides by John Loucks St. Edward’s University

2 2 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 6 Continuous Probability Distributions n Uniform Probability Distribution f ( x ) x x Uniform x Normal x x Exponential n Normal Probability Distribution n Normal Approximation of Binomial Probabilities n Exponential Probability Distribution

3 3 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

4 4 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Continuous Probability Distributions 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. f ( x ) x x Uniform x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 x Normal x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 Exponential x x x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2

5 5 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Uniform Probability Distribution where: a = smallest value the variable can assume b = largest value the variable can assume b = largest value the variable can assume f ( x ) = 1/( b – a ) for a < x < b f ( x ) = 1/( b – a ) for a < x < b = 0 elsewhere = 0 elsewhere f ( x ) = 1/( b – a ) for a < x < b f ( x ) = 1/( b – a ) for a < x < b = 0 elsewhere = 0 elsewhere n A random variable is uniformly distributed whenever the probability is proportional to the interval’s length. n The uniform probability density function is:

6 6 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Var( x ) = ( b - a ) 2 /12 E( x ) = ( a + b )/2 Uniform Probability Distribution n Expected Value of x n Variance of x

7 7 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Uniform Probability Distribution n Example: Slater's Buffet Slater customers are charged for the amount of 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.

8 8 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Uniform Probability Density Function f ( x ) = 1/10 for 5 < x < 15 f ( x ) = 1/10 for 5 < x < 15 = 0 elsewhere = 0 elsewhere f ( x ) = 1/10 for 5 < x < 15 f ( x ) = 1/10 for 5 < x < 15 = 0 elsewhere = 0 elsewhere where: x = salad plate filling weight x = salad plate filling weight Uniform Probability Distribution

9 9 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Expected Value of x E( x ) = ( a + b )/2 E( x ) = ( a + b )/2 = (5 + 15)/2 = (5 + 15)/2 = 10 = 10 E( x ) = ( a + b )/2 E( x ) = ( a + b )/2 = (5 + 15)/2 = (5 + 15)/2 = 10 = 10 Var( x ) = ( b - a ) 2 /12 Var( x ) = ( b - a ) 2 /12 = (15 – 5) 2 /12 = (15 – 5) 2 /12 = 8.33 = 8.33 Var( x ) = ( b - a ) 2 /12 Var( x ) = ( b - a ) 2 /12 = (15 – 5) 2 /12 = (15 – 5) 2 /12 = 8.33 = 8.33 Uniform Probability Distribution n Variance of x

10 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Uniform Probability Distribution for Salad Plate Filling Weight f(x)f(x) f(x)f(x) x x 1/10 Salad Weight (oz.) Uniform Probability Distribution 5 5 10 15 0 0

11 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. f(x)f(x) f(x)f(x) x x 1/10 Salad Weight (oz.) 5 5 10 15 0 0 P(12 < x < 15) = 1/10(3) =.3 What is the probability that a customer What is the probability that a customer will take between 12 and 15 ounces of salad? will take between 12 and 15 ounces of salad? Uniform Probability Distribution 12

Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Area as a Measure of Probability n The area under the graph of f ( x ) and probability are identical. n This is valid for all continuous random variables. n The probability that x takes on a value between some lower value x 1 and some higher value x 2 can be found by computing the area under the graph of f ( x ) over the interval from x 1 to x 2.

13 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Normal Probability Distribution n The normal probability distribution is the most important distribution for describing a continuous random variable. n It is widely used in statistical inference. n It has been used in a wide variety of applications including: including: Heights of people Heights of people Rainfall amounts Rainfall amounts Test scores Test scores Scientific measurements Scientific measurements n Abraham de Moivre, a French mathematician, published The Doctrine of Chances in 1733. n He derived the normal distribution.

14 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Normal Probability Distribution n Normal Probability Density Function  = mean  = standard deviation  = 3.14159 e = 2.71828 where:

15 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The distribution is symmetric; its skewness The distribution is symmetric; its skewness measure is zero. measure is zero. The distribution is symmetric; its skewness The distribution is symmetric; its skewness measure is zero. measure is zero. Normal Probability Distribution n Characteristics x

16 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The entire family of normal probability The entire family of normal probability distributions is defined by its mean  and its distributions is defined by its mean  and its standard deviation . standard deviation . The entire family of normal probability The entire family of normal probability distributions is defined by its mean  and its distributions is defined by its mean  and its standard deviation . standard deviation . Normal Probability Distribution n Characteristics Standard Deviation  Mean  x

17 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The highest point on the normal curve is at the The highest point on the normal curve is at the mean, which is also the median and mode. mean, which is also the median and mode. The highest point on the normal curve is at the The highest point on the normal curve is at the mean, which is also the median and mode. mean, which is also the median and mode. Normal Probability Distribution n Characteristics x

18 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Normal Probability Distribution n Characteristics -10025 The mean can be any numerical value: negative, The mean can be any numerical value: negative, zero, or positive. zero, or positive. The mean can be any numerical value: negative, The mean can be any numerical value: negative, zero, or positive. zero, or positive. x

19 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Normal Probability Distribution n Characteristics  = 15  = 25 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. x

20 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Probabilities for the normal random variable are Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). Probabilities for the normal random variable are Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). Normal Probability Distribution n Characteristics.5.5 x

21 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Normal Probability Distribution n Characteristics (basis for the empirical rule) of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean.68.26%68.26% +/- 1 standard deviation of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. 95.44%95.44% +/- 2 standard deviations of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean.99.72%99.72% +/- 3 standard deviations

22 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Normal Probability Distribution n Characteristics (basis for the empirical rule) x  – 3   – 1   – 2   + 1   + 2   + 3  68.26% 95.44% 99.72%

23 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Standard Normal Probability Distribution A random variable having a normal distribution A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is with a mean of 0 and a standard deviation of 1 is said to have a standard normal probability said to have a standard normal probability distribution. distribution. A random variable having a normal distribution A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is with a mean of 0 and a standard deviation of 1 is said to have a standard normal probability said to have a standard normal probability distribution. distribution. n Characteristics

24 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part.  0 z The letter z is used to designate the standard The letter z is used to designate the standard normal random variable. normal random variable. The letter z is used to designate the standard The letter z is used to designate the standard normal random variable. normal random variable. Standard Normal Probability Distribution n Characteristics

25 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 .

26 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Standard Normal Probability Distribution n Example: Pep Zone Pep Zone sells auto parts and supplies including 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 The store manager is concerned that sales are being lost due to stockouts while waiting for a replenishment order.

27 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. It has been determined that demand during 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. Standard Normal Probability Distribution n Example: Pep Zone The manager would like to know the probability The manager would like to know the probability of a stockout during replenishment lead-time. In other words, what is the probability that demand during lead-time will exceed 20 gallons? P ( x > 20) = ? P ( x > 20) = ?

28 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. z = ( x -  )/  z = ( x -  )/  = (20 - 15)/6 = (20 - 15)/6 =.83 =.83 z = ( x -  )/  z = ( x -  )/  = (20 - 15)/6 = (20 - 15)/6 =.83 =.83 n Solving for the Stockout Probability Step 1: Convert x to the standard normal distribution. 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

29 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Cumulative Probability Table for the Standard Normal Distribution P ( z <.83) Standard Normal Probability Distribution

30 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. P ( z >.83) = 1 – P ( z.83) = 1 – P ( z <.83) = 1-.7967 = 1-.7967 =.2033 =.2033 P ( z >.83) = 1 – P ( z.83) = 1 – P ( z <.83) = 1-.7967 = 1-.7967 =.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. Probability of a stockout of a stockout P ( x > 20) Standard Normal Probability Distribution

31 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Solving for the Stockout Probability 0.83 Area =.7967 Area = 1 -.7967 =.2033 =.2033 z Standard Normal Probability Distribution

32 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Standard Normal Probability Distribution Standard Normal Probability Distribution If the manager of Pep Zone wants the probability If the manager of Pep Zone wants the probability of a stockout during replenishment lead-time to be no more than.05, what should the reorder point be? --------------------------------------------------------------- --------------------------------------------------------------- (Hint: Given a probability, we can use the standard normal table in an inverse fashion to find the corresponding z value.)

33 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Solving for the Reorder Point 0 Area =.9500 Area =.0500 z z.05 Standard Normal Probability Distribution

34 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Solving for the Reorder Point 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 (1 -.05 =.95) Standard Normal Probability Distribution

35 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Solving for the Reorder Point Step 2: Convert z.05 to the corresponding value of x. x =  + z.05  x =  + z.05   = 15 + 1.645(6) = 24.87 or 25 = 24.87 or 25 x =  + z.05  x =  + z.05   = 15 + 1.645(6) = 24.87 or 25 = 24.87 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

36 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Normal Probability Distribution n Solving for the Reorder Point 15 x 24.87 Probability of a stockout during replenishment lead-time =.05 Probability of a stockout during replenishment lead-time =.05 Probability of no stockout during replenishment lead-time =.95 Probability of no stockout during replenishment lead-time =.95

37 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Solving for the Reorder Point 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 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

38 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 evaluating the binomial probability function by hand or with a calculator is difficult. or with a calculator is difficult. When the number of trials, n, becomes large, When the number of trials, n, becomes large, evaluating the binomial probability function by hand evaluating the binomial probability function by hand or with a calculator is difficult. 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 easy-to-use approximation of binomial probabilities where np > 5 and n(1 - p) > 5. where np > 5 and n(1 - p) > 5. The normal probability distribution provides an The normal probability distribution provides an easy-to-use approximation of binomial probabilities easy-to-use approximation of binomial probabilities where np > 5 and n(1 - p) > 5. where np > 5 and n(1 - p) > 5. In the definition of the normal curve, set In the definition of the normal curve, set  = np and In the definition of the normal curve, set In the definition of the normal curve, set  = np and

39 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Add and subtract a continuity correction factor Add and subtract a continuity correction factor because a continuous distribution is being used to because a continuous distribution is being used to approximate a discrete distribution. approximate a discrete distribution. Add and subtract a continuity correction factor Add and subtract a continuity correction factor because a continuous distribution is being used to because a continuous distribution is being used to approximate a discrete distribution. approximate a discrete distribution. For example, P ( x = 12) for the discrete binomial For example, P ( x = 12) for the discrete binomial probability distribution is approximated by probability distribution is approximated by P (11.5 < x < 12.5) for the continuous normal P (11.5 < x < 12.5) for the continuous normal distribution. distribution. For example, P ( x = 12) for the discrete binomial For example, P ( x = 12) for the discrete binomial probability distribution is approximated by probability distribution is approximated by P (11.5 < x < 12.5) for the continuous normal P (11.5 < x < 12.5) for the continuous normal distribution. distribution. Normal Approximation of Binomial Probabilities

40 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Normal Approximation of Binomial Probabilities n Example Suppose that a company has a history of making errors in 10% of its invoices. A sample of 100 invoices has been taken, and we want to compute the probability that 12 invoices contain errors. Suppose that a company has a history of making errors in 10% of its invoices. A sample of 100 invoices has been taken, and we want to compute the probability that 12 invoices contain errors. In this case, we want to find the binomial probability of 12 successes in 100 trials. So, we set: In this case, we want to find the binomial probability of 12 successes in 100 trials. So, we set:  = np = 100(.1) = 10  = np = 100(.1) = 10 = [100(.1)(.9)] ½ = 3 = [100(.1)(.9)] ½ = 3

41 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Normal Approximation of Binomial Probabilities n Normal Approximation to a Binomial Probability Distribution with n = 100 and p =.1 Distribution with n = 100 and p =.1  = 10 P (11.5 < x < 12.5) (Probability (Probability of 12 Errors) of 12 Errors) x 11.5 12.5  = 3

42 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Normal Approximation to a Binomial Probability Distribution with n = 100 and p =.1 Distribution with n = 100 and p =.1 10 P ( x < 12.5) =.7967 x 12.5 Normal Approximation of Binomial Probabilities

43 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Normal Approximation of Binomial Probabilities n Normal Approximation to a Binomial Probability Distribution with n = 100 and p =.1 Distribution with n = 100 and p =.1 10 P ( x < 11.5) =.6915 x 11.5

44 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Normal Approximation of Binomial Probabilities 10 P ( x = 12) =.7967 -.6915 =.7967 -.6915 =.1052 =.1052 x 11.5 12.5 n The Normal Approximation to the Probability of 12 Successes in 100 Trials is.1052 of 12 Successes in 100 Trials is.1052

45 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Exponential Probability Distribution n The exponential probability distribution is useful in describing the time it takes to complete a task. 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 n The exponential random variables can be used to describe: n In waiting line applications, the exponential distribution is often used for service times.

46 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Exponential Probability Distribution n A property of the exponential distribution is that the mean and standard deviation are equal. n The exponential distribution is skewed to the right. Its skewness measure is 2.

47 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Density Function Exponential Probability Distribution where:  = expected or mean e = 2.71828 e = 2.71828 for x > 0

48 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Cumulative Probabilities Exponential Probability Distribution where: x 0 = some specific value of x x 0 = some specific value of x

49 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Exponential Probability Distribution n Example: Al’s Full-Service Pump The time between arrivals of cars at Al’s full- 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.

50 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. x x f(x)f(x) f(x)f(x).1.3.4.2 0 1 2 3 4 5 6 7 8 9 10 Time Between Successive Arrivals (mins.) Exponential Probability Distribution P ( x < 2) = 1 - 2.71828 -2/3 = 1 -.5134 =.4866 P ( x < 2) = 1 - 2.71828 -2/3 = 1 -.5134 =.4866 n Example: Al’s Full-Service Pump

51 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

52 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 6