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part © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in Business Analytics: Data Analysis and Chapter Decision Making Normal, Binomial, Poisson, and Exponential Distributions Introduction Several specific distributions commonly occur in a variety of business situations: Normal distribution—a continuous distribution characterized by a symmetric bell-shaped curve Binomial distribution—a discrete distribution that is relevant when we sample from a population with only two types of members or when we perform a series of independent, identical experiments with only two possible outcomes Poisson distribution—a discrete distribution that describes the number of events in any period of time Exponential distributions—a continuous distribution that describes the times between events © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part The Normal Distribution The single most important distribution in statistics is the normal distribution It is a continuous distribution and is the basis of the familiar symmetric bellshaped curve Any particular normal distribution is specified by its mean and standard deviation By changing the mean, the normal curve shifts to the right or left By changing the standard deviation, the curve becomes more or less spread out There are really many normal distributions, not just a single one The normal distribution is a two-parameter family, where the two parameters are the mean and standard deviation © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Continuous Distributions and Density Functions (slide of 2) For continuous distributions, instead of a list of possible values, there is a continuum of possible values, such as all values between and 100 or all values greater than Instead of assigning probabilities to each individual value in the continuum, the total probability of is spread over this continuum The key to this spreading is called a density function, which acts like a histogram The higher the value of the density function, the more likely this region of the continuum is © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Continuous Distributions and Density Functions (slide of 2) A density function, usually denoted by f(x), specifies the probability distribution of a continuous random variable X The higher f(x) is, the more likely x is The total area between the graph of f(x) and the horizontal axis, which represents the total probability, is equal to f(x) is nonnegative for all possible values of X Probabilities are found from a density function as areas under the curve © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part The Normal Density The normal distribution is a continuous distribution with possible values ranging over the entire number line—from “minus infinity” to “plus infinity.” Only a relatively small range has much chance of occurring The normal density function is actually quite complex, in spite of its “nice” bell-shaped appearance The formula for the normal density function, where μ and σ are the mean and standard deviation, is: © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Standardizing: Z-Values The standard normal distribution has mean and standard deviation 1, so it is denoted by N(0,1) It is also referred to as the Z distribution To standardize a variable, subtract its mean and then divide the difference by the standard deviation: A Z-value is the number of standard deviations to the right or left of the mean If Z is positive, the original value is to the right of the mean If Z is negative, the original value is the left of the mean © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 5.1: Standardizing.xlsx Objective: To use Excel® to standardize annual returns of various mutual funds Solution: Data set includes the annual returns of 30 mutual funds Calculate the mean and standard deviation of each annual return and then use the standardizing formula to calculate the corresponding Zvalue OR calculate the Z-values directly, using Excel’s STANDARDIZE function © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Normal Tables and Z-Values A common use for Z-values and the standard normal distribution is in calculating probabilities and percentiles by the traditional method This method is based on a table of the standard normal distribution found in many statistics textbooks An example of such a table is given below The body of the table contains probabilities The left and top margins contain possible values © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Normal Calculations in Excel Two types of calculations are typically made with normal distributions: finding probabilities and finding percentiles The functions used for normal probability calculations are NORMDIST and NORMSDIST The main difference between these is that the one with the “S” (for standardized) applies only to N(0, 1) calculations, whereas NORMDIST applies to any normal distribution Percentile calculations that take a probability and return a value are often called inverse calculations The Excel functions for these are named NORMINV and NORMSINV Again, the “S” in the second of these indicates that it applies to the standard normal distribution © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part The Normal Approximation to the Binomial If you graph the binomial probabilities, you will see an interesting phenomenon: the graph begins to look symmetric and bell-shaped when n is fairly large and p is not too close to or The normal distribution provides a very good approximation to the binomial under these conditions One practical consequence of the normal approximation to the binomial is that the empirical rules apply very well to binomial distributions © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 5.8: Beating the Market.xlsx Objective: To determine the probability of a mutual fund outperforming a standard market index at least 37 out of 52 weeks Solution: The number of weeks where a given fund outperforms the market index is binomially distributed with n = 52 and p = 0.5 This probability is quite small (0.00159) Now let Y be the number of the 400 best mutual funds that beat the market at least 37 of 52 weeks Y is also binomially distributed, with parameters n = 400 and p = 0.00159 The resulting probability is nearly 0.5 © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 5.9: Supermarket Spending.xlsx Objective: To use the normal and binomial distributions to calculate the typical number of customers who spend at least $100 per day and the probability that at least 30% of all 500 daily customers spend at least $100 Solution: Historical data indicate that the amount spent per customer is normally distributed with mean $85 and standard deviation $30 If 500 customers shop in a given day, calculate the mean and standard deviation of the number who spend at least $100 Then calculate the probability that at least 30% of the 500 customers spend at least $100 This is the probability that a binomially distributed random variable, with n = 500 and p = 0.309, is at least 150 © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 5.10: Airline Overbooking.xlsx (slide of 2) Objective: To assess the benefits and drawbacks of airline overbooking Solution: Assume that the no-show rate is 10%—that is, each ticketed passenger shows up with probability 0.90 For a flight with 200 seats, calculate the probability that more than 205 passengers show up; that more than 200 passengers show up; that at least 195 seats are filled; and that at least 190 seats are filled Use the BINOMDIST function and a data table to determine the probabilities © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 5.10: Airline Overbooking.xlsx (slide of 2) To see how sensitive these probabilities are to the number of tickets issued, create a one-way data table, as shown at the bottom of the spreadsheet below © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 5.11: Election Returns.xlsx Objective: To use a binomial model to determine whether early returns reflect the eventual winner of an election between two candidates Solution: Suppose that a small percentage of the votes have been counted and the Republican is currently ahead 540 to 460 On what basis can the networks declare the Republican the winner, if there are millions of voters? Use a binomial model to see how unlikely the event “at least 540 out of 1000” is, assuming that the Democrat will be the eventual winner © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 5.12: Basketball Simulation.xlsx Objective: To formulate a nonbinomial model of basketball shooting, and to use it to find the probability of a “450 shooter” making at least 13 out of 25 shots Solution: Assume the shooter makes 45% of his shots in the long run Use simulation to create a model that implies that the shooter gets better the more shots he makes and worse the more he misses Consider his nth shot If he has made his last k shots, assume the probability of making shot n is 0.45 + kd1 If he has missed his last k shots, assume the probability of making shot n is 0.45 − kd2 © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part The Poisson and Exponential Distributions In most statistical applications, the Poisson and exponential distributions play a much less important role than the normal and binomial distributions However, in many applied management science models, the Poisson and exponential distributions are key distributions For example, much of the study of probabilistic inventory models, queuing models, and reliability models relies heavily on these two distributions © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part The Poisson Distribution (slide of 3) The Poisson distribution is a discrete distribution It usually applies to the number of events occurring within a specified period of time or space Its possible values are all of the nonnegative integers: 0, 1, 2, and so on— there is no upper limit Even though there is an infinite number of possible values, this causes no real problems because the probabilities of all sufficiently large values are essentially The Poisson distribution is characterized by a single parameter, usually labeled λ (Greek lambda), which must be positive It is both the mean and the variance of the Poisson distribution It is often called a rate—arrivals per hour, for example © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part The Poisson Distribution (slide of 3) All Poisson distributions have the same basic shape as in the figure below That is, they first increase and then decrease © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part The Poisson Distribution (slide of 3) Typical examples of the Poisson distribution: A bank manager is studying the arrival pattern to the bank The events are customer arrivals, the number of arrivals in an hour is Poisson distributed, and λ represents the expected number of arrivals per hour A retailer is interested in the number of customers who order a particular product in a week Then the events are customer orders for the product, the number of customer orders in a week is Poisson distributed, and λ is the expected number of orders per week In Excel, calculate Poisson probabilities with the POISSON function © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 5.13: Poisson Demand Distribution.xlsx (slide of 2) Objective: To model the probability distribution of monthly demand for plasma screen TVs with a particular Poisson distribution Solution: Because the histogram of demands from previous months resembles a Poisson distribution, try modeling the monthly demand with a Poisson distribution The historical average demand per month is about 17, so let the mean demand per month λ = 17 Now test the Poisson model by calculating the probabilities of various events © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 5.13: Poisson Demand Distribution.xlsx (slide of 2) © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part The Exponential Distribution (slide of 2) The most common probability distribution used to model the times between customer arrivals, often called interarrival times, is the exponential distribution In general, the continuous random variable X has an exponential distribution with parameter λ (with λ > 0) if the density function of X has the form: The mean and standard deviation of this distribution are both equal to the reciprocal of the parameter λ For any exponential distribution, the probability to the left of a given value x > can be calculated with Excel’s EXPONDIST function © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part The Exponential Distribution (slide of 2) The exponential density function has the shape shown below Because this density function decreases continuously from left to right, its most likely value is x = © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part ... mean Almost all fall within three standard deviations of the mean For these rules to hold with real data, the distribution of the data must be at least approximately symmetric and bell-shaped... duplicated, or posted to a publicly accessible website, in whole or in part Example 5.12: Basketball Simulation.xlsx Objective: To formulate a nonbinomial model of basketball shooting, and. .. (for standardized) applies only to N(0, 1) calculations, whereas NORMDIST applies to any normal distribution Percentile calculations that take a probability and return a value are often called
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