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 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 For a discrete random variable, the probability function f(x) provides the probability that the random variable assumes a particular value. With continuous random variables, the counterpart of the probability function is the probability density function, also f(x). The difference is the probability density function does not directly provide probabilities.

4 4 Slide © 2008 Thomson South-Western. All Rights Reserved n What observations can you make about the area under the graph of f(x) and probability? n They are identical! This observation is valid for all continuous random variables. n Once a probability density function f(x) is identified, the probability that x takes 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 x1 to x2. n The total area under the graph of f(x) for continuous probability distributions is equal to 1. This property is the analog of the condition that the sum of the probabilities must equal 1 for discrete probability functions. Continuous Probability Distributions

5 5 Slide © 2008 Thomson South-Western. All Rights Reserved 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

6 6 Slide © 2008 Thomson South-Western. All Rights Reserved 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:

7 7 Slide © 2008 Thomson South-Western. All Rights Reserved Var( x ) = ( b - a ) 2 /12 E( x ) = ( a + b )/2 Uniform Probability Distribution n Expected Value of x n Variance of x

8 8 Slide © 2008 Thomson South-Western. All Rights Reserved Uniform Probability Distribution n Example: Slater's Buffet Slater’s 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.

9 9 Slide © 2008 Thomson South-Western. All Rights Reserved 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

10 Slide © 2008 Thomson South-Western. All Rights Reserved n Expected Value of x n Variance 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

11 Slide © 2008 Thomson South-Western. All Rights Reserved n Uniform Probability Distribution for Salad Plate Filling Weight f(x)f(x) f(x)f(x) x x /10 Salad Weight (oz.) Uniform Probability Distribution

12 Slide © 2008 Thomson South-Western. All Rights Reserved f(x)f(x) f(x)f(x) x x /10 Salad Weight (oz.) 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? 12 Uniform Probability Distribution

13 Slide © 2008 Thomson South-Western. All Rights Reserved 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.

14 Slide © 2008 Thomson South-Western. All Rights Reserved Heights of people Heights Normal Probability Distribution n It has been used in a wide variety of applications: Scientific measurements measurementsScientific

15 Slide © 2008 Thomson South-Western. All Rights Reserved Amounts of rainfall Amounts Normal Probability Distribution n It has been used in a wide variety of applications: Test scores scoresTest

16 Slide © 2008 Thomson South-Western. All Rights Reserved Normal Probability Distribution n Normal Probability Density Function  = mean  = standard deviation  = e = where:

17 Slide © 2008 Thomson South-Western. All Rights Reserved 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

18 Slide © 2008 Thomson South-Western. All Rights Reserved 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

19 Slide © 2008 Thomson South-Western. All Rights Reserved 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 In theory, the normal distribution is perfectly In theory, the normal distribution is perfectly symmetrical. The tails extend to infinity and never symmetrical. The tails extend to infinity and never touch the horizontal axis. It is not skewed and its touch the horizontal axis. It is not skewed and its skewness measure is zero. However, real world skewness measure is zero. However, real world occurrences deviate from perfectly symmetrical. occurrences deviate from perfectly symmetrical. In theory, the normal distribution is perfectly In theory, the normal distribution is perfectly symmetrical. The tails extend to infinity and never symmetrical. The tails extend to infinity and never touch the horizontal axis. It is not skewed and its touch the horizontal axis. It is not skewed and its skewness measure is zero. However, real world skewness measure is zero. However, real world occurrences deviate from perfectly symmetrical. occurrences deviate from perfectly symmetrical.

20 Slide © 2008 Thomson South-Western. All Rights Reserved Normal Probability Distribution n Characteristics 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

21 Slide © 2008 Thomson South-Western. All Rights Reserved 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

22 Slide © 2008 Thomson South-Western. All Rights Reserved 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

23 Slide © 2008 Thomson South-Western. All Rights Reserved 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% +/- 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

24 Slide © 2008 Thomson South-Western. All Rights Reserved Normal Probability Distribution n Characteristics x  – 3   – 1   – 2   + 1   + 2   + 3  68.26% 95.44% 99.72%

25 Slide © 2008 Thomson South-Western. All Rights Reserved 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. The three types of probabilities we need to compute The three types of probabilities we need to compute include: (1) the probability that the standard normal include: (1) the probability that the standard normal random variable z will be less than or equal to a random variable z will be less than or equal to a given value; (2) the probability that z will be given value; (2) the probability that z will be between two given values; and (3) the probability between two given values; and (3) the probability that z will be greater than or equal to given value. that z will be greater than or equal to given value. The three types of probabilities we need to compute The three types of probabilities we need to compute include: (1) the probability that the standard normal include: (1) the probability that the standard normal random variable z will be less than or equal to a random variable z will be less than or equal to a given value; (2) the probability that z will be given value; (2) the probability that z will be between two given values; and (3) the probability between two given values; and (3) the probability that z will be greater than or equal to given value. that z will be greater than or equal to given value.

26 Slide © 2008 Thomson South-Western. All Rights Reserved Types of Probabilities We Need To Compute z 1 z 2 z 3 z 1 z 2 z 3 Type #1 Type #3 Type #2 z=0 z=1

27 Slide © 2008 Thomson South-Western. All Rights Reserved Types of Questions We Will Ask The first type of question specifies a value, or The first type of question specifies a value, or values, for z and asks us to use the table for values, for z and asks us to use the table for cumulative probabilities for the standard normal cumulative probabilities for the standard normal distribution to determine the corresponding areas distribution to determine the corresponding areas or probabilities. or probabilities. The first type of question specifies a value, or The first type of question specifies a value, or values, for z and asks us to use the table for values, for z and asks us to use the table for cumulative probabilities for the standard normal cumulative probabilities for the standard normal distribution to determine the corresponding areas distribution to determine the corresponding areas or probabilities. or probabilities. The second type of question provides an area, or The second type of question provides an area, or probability, and asks us to use the table for probability, and asks us to use the table for cumulative probabilities for the standard normal cumulative probabilities for the standard normal distribution to determine the corresponding z distribution to determine the corresponding z value. value. The second type of question provides an area, or The second type of question provides an area, or probability, and asks us to use the table for probability, and asks us to use the table for cumulative probabilities for the standard normal cumulative probabilities for the standard normal distribution to determine the corresponding z distribution to determine the corresponding z value. value. Sketching a graph of the standard normal Sketching a graph of the standard normal probability distribution and shading the probability distribution and shading the appropriate area will help to define the question. appropriate area will help to define the question. Sketching a graph of the standard normal Sketching a graph of the standard normal probability distribution and shading the probability distribution and shading the appropriate area will help to define the question. appropriate area will help to define the question.

28 Slide © 2008 Thomson South-Western. All Rights Reserved  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

29 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 .

30 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

31 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). This is a Type #3 case. Type #3 case. Standard Normal Probability Distribution Pep Zone 5w-20 Motor Oil n Example: Pep Zone

32 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

33 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

34 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

35 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

36 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

37 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

38 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

39 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

40 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

41 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.

42 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).

43 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 or distance between occurrences of interest. 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

44 Slide © 2008 Thomson South-Western. All Rights Reserved n Density Function Exponential Probability Distribution where:  = mean e = e = for x > 0,  > 0 n The exponential probability distribution provides the cumulative probability of obtaining a value for the exponential random variable of less than or equal to some specific value denoted by x.

45 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 n As with any continuous probability distribution, the area under the curve corresponding to an interval provides the probability that the random variable assumes a value in that interval.

46 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.

47 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

48 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.

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