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Chapter 3 part B Probability Distribution

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Chapter 3, Part B Probability Distributions n Uniform Probability Distribution n Normal Probability Distribution n Exponential Probability Distribution f ( x ) x x Uniform x Normal x x Exponential

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Continuous Random Variables n Examples of continuous random variables include the following: The number of ounces of soup placed in a can The number of ounces of soup placed in a can The flight time of an airplane traveling from Chicago to New York The flight time of an airplane traveling from Chicago to New York The lifetime of the picture tube in a new television set The lifetime of the picture tube in a new television set The drilling depth required to reach oil in an offshore drilling operation The drilling depth required to reach oil in an offshore drilling operation

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Continuous Probability Distributions n A continuous random variable can assume any value in an interval on the real. n. 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 an. n Instead, we talk about the probability of the random variable assuming a value within an interval.

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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

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n A random variable is uniformly distributed whenever the probability that the variable will assume a value in any interval of equal length is the same for each interval. n The uniform probability density function is: 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

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Example: Flight Time n Uniform Probability Distribution Let x denote the flight time of an airplane traveling from Chicago to New York. Assume that the minimum time is 2 hours and that the maximum time is 2 hours 20 minutes. n Assume that sufficient actual flight data are available to conclude that the probability of a flight time is same in this interval. n This means probability of flight time between 120 and 121 minutes is the same as the probability of a flight time within any other 1-minute interval up to and including 140 minutes.

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n Uniform Probability Density Function Example: Flight Time f ( x ) = 1/20 for 120 < x < 140 f ( x ) = 1/20 for 120 < x < 140 = 0 elsewhere = 0 elsewhere f ( x ) = 1/20 for 120 < x < 140 f ( x ) = 1/20 for 120 < x < 140 = 0 elsewhere = 0 elsewhere where: x = flight time in minutes x = flight time in minutes

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n Uniform Probability Distribution for Flight Time f(x)f(x) f(x)f(x) x x 120 130 140 1/20 Flight Time (mins.) Example: Flight Time

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f(x)f(x) f(x)f(x) x x 120 130 140 1/20 Flight Time (mins.) Example: Flight Time P(135 < x < 140) = 1/20(5) =.25 What is the probability that a flight will take What is the probability that a flight will take between 135 and 140 minutes? 135

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f(x)f(x) f(x)f(x) x x 120 130 140 1/20 Flight Time (mins.) Example: Flight Time P(124 < x < 136) = 1/20(12) =.6 What is the probability that a flight will take What is the probability that a flight will take between 124 and 136 minutes? 136 124

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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.

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Heights of people Heights Normal Probability Distribution n It has been used in a wide variety of applications: Scientific measurements measurementsScientific Test scores scoresTest Amounts of rainfall Amounts

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Normal Probability Distribution n Normal Probability Density Function = mean = standard deviation = 3.14159 e = 2.71828 where:

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The distribution is symmetric, and is bell-shaped. The distribution is symmetric, and is bell-shaped. Normal Probability Distribution n Characteristics x

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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

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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

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Normal Probability Distribution n Characteristics -10020 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

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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

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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

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Normal Probability Distribution n Characteristics 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

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Normal Probability Distribution n Characteristics x – 3 – 1 – 2 + 1 + 2 + 3 68.26% 95.44% 99.72%

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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.

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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

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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 .

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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.

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Example: Pep Zone n Standard Normal Probability Distribution 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).

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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 Example: Pep Zone Step 1: Convert x to the standard normal distribution. Step 1: Convert x to the standard normal distribution. Step 2: Find the area under the standard normal Step 2: Find the area under the standard normal curve between the mean and z =.83. curve between the mean and z =.83. Step 2: Find the area under the standard normal Step 2: Find the area under the standard normal curve between the mean and z =.83. curve between the mean and z =.83. see next slide see next slide

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n Probability Table for the Standard Normal Distribution Example: Pep Zone P (0 < z <.83)

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P ( z >.83) =.5 – P (0.83) =.5 – P (0 < z <.83) = 1-.2967 = 1-.2967 =.2033 =.2033 P ( z >.83) =.5 – P (0.83) =.5 – P (0 < z <.83) = 1-.2967 = 1-.2967 =.2033 =.2033 n Solving for the Stockout Probability Example: Pep Zone Step 3: Compute the area under the standard normal 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 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 stockoutProbability P ( x > 20)

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n Solving for the Stockout Probability Example: Pep Zone 0.83 Area =.2967 Area =.5 -.2967 =.2033 =.2033 z

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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? Example: Pep Zone

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n Solving for the Reorder Point 0 Area =.4500 Area =.0500 z z.05 Example: Pep Zone

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n Solving for the Reorder Point Example: Pep Zone Step 1: Find the z -value that cuts off an area of.05 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 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 area (.5 -.05 =.45) We look up the area (.5 -.05 =.45)

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n Solving for the Reorder Point Example: Pep Zone Step 2: Convert z.05 to the corresponding value of x. 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.

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n Solving for the Reorder Point Example: Pep Zone 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.

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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

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n Density Function Exponential Probability Distribution where: = mean e = 2.71828 e = 2.71828 for x > 0, > 0

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n Cumulative Probabilities Exponential Probability Distribution where: x 0 = some specific value of x x 0 = some specific value of x

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n Exponential Probability Distribution 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. Example: Al’s Full-Service Pump

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n Exponential Probability Distribution x x f(x)f(x) f(x)f(x).1.3.4.2 1 2 3 4 5 6 7 8 9 10 Time Between Successive Arrivals (mins.) Example: Al’s Full-Service Pump P ( x < 2) = 1 - 2.71828 -2/3 = 1 -.5134 =.4866 P ( x < 2) = 1 - 2.71828 -2/3 = 1 -.5134 =.4866

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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

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End of Chapter 3, Part B

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