MATH112-WEEK 7-LESSON 13-Discrete Distributions.pdf

STATISTICAL ANALYSIS WITH SOFTWARE APPLICATION Discrete Distributions MODULE 2-WEEK 7-LESSON 13 Excellence and Relevance REMELYN L. ASA...

STATISTICAL ANALYSIS WITH SOFTWARE APPLICATION Discrete Distributions MODULE 2-WEEK 7-LESSON 13 Excellence and Relevance REMELYN L. ASAHID-CHENG Discrete Random Variable A random variable is a discrete random variable if the set of all possible values is at most a finite or a countably infinite number of possible values. In most statistical situations, discrete random variables produce values that are non-negative whole numbers. For example, if six people are randomly selected from a population and how many of the six are left-handed is to be determined, the random variable produced is discrete. The only possible numbers of left-handed people in the sample of six are 0, 1, 2, 3, 4, 5, and 6. There cannot be 2.75 left-handed people in a group of six people; obtaining nonwhole number values is impossible. Other examples of experiments that yield discrete random variables include the following. 1. Randomly selecting 25 people who consume soft drinks and determining how many people prefer diet soft drinks 2. Determining the number of defective items in a batch of 50 items 3. Counting the number of people who arrive at a store during a 5-minute period 4. Sampling 100 registered voters and determining how many voted for the president in the last election Excellence and Relevance Describing a Discrete Distribution How can we describe a discrete distribution? One way is to construct a graph of the distribution and study the graph. The histogram is probably the most common graphical way to depict a discrete distribution. Observe the discrete distribution in Table 5.2. An executive is considering out-of-town business travel for a given Friday. She recognizes that at least one crisis could occur on the day that she is gone and she is concerned about that possibility. Table 5.2 shows a discrete distribution that contains the number of crises that could occur during the day that she is gone and the probability that each number will occur. For example, there is a.37 probability that no crisis will occur, a.31 probability of one crisis, and so on. The histogram in Figure 5.1 depicts the distribution given in Table 5.2. Notice that the x-axis of the histogram contains the possible outcomes of the experiment (number of crises that might occur) and that the y-axis contains the probabilities of these occurring. j Excellence and Relevance Mean of Discrete Distributions The mean or expected value of a discrete distribution is the long-run average of occurrences. We must realize that any one trial using a discrete random variable yields only one outcome. However, if the process is repeated long enough, the average of the outcomes is most likely to approach a long-run average, expected value, or mean value. This mean, or expected, value is computed as follows. Excellence and Relevance Mean of Discrete Distributions As an example, let’s compute the mean or expected value of the distribution given in Table 5.2. See Table 5.3 for the resulting values. In the long run, the mean or expected number of crises on a given Friday for this executive is 1.15 crises. Of course, the executive will never have 1.15 crises. Excellence and Relevance Variance and Standard Deviation of a Discrete Distribution The variance and standard deviation of a discrete distribution are solved for by using the outcomes (x) and probabilities of outcomes [P(x)] in a manner similar to that of computing a mean. In addition, the computations for variance and standard deviations use the mean of the discrete distribution. The formula for computing the variance follows Excellence and Relevance Variance and Standard Deviation of a Discrete Distribution The variance and standard deviation of the crisis data in Table 5.2 are calculated and shown in Table 5.4. The mean of the crisis data is 1.15 crises. The standard deviation is 1.19 crises, and the variance is 1.41. Excellence and Relevance Demonstration Problem During one holiday season, the Texas lottery played a game called the Stocking Stuffer. With this game, total instant winnings of $34.8 million were available in 70 million $1 tickets, with ticket prizes ranging from $1 to $1,000. Shown here are the various prizes and the probability of winning each prize. Use these data to compute the expected value of the game, the variance of the game, and the standard deviation of the game. Solution The mean is computed as follows. Excellence and Relevance Demonstration Problem The expected payoff for a $1 ticket in this game is 60.2 cents. If a person plays the game for a long time, he or she could expect to average about 60 cents in winnings. In the long run, the participant will lose about $1.00 −.602 =.398, or about 40 cents a game. Of course, an individual will never win 60 cents in any one game. Using this mean, μ =.60155, the variance and standard deviation can be computed as follows. The variance is 28.98349 (dollars)2 and the standard deviation is $5.38. Excellence and Relevance  Binomial Distribution Excellence and Relevance Binomial Distribution Perhaps the most widely known of all discrete distributions is the binomial distribution. The binomial distribution has been used for hundreds of years. Besides its historical significance, the binomial distribution is studied by business statistics students because it is the basis for other important statistical techniques. In studying various distributions, it is important to be able to differentiate among distributions and to know the unique characteristics and assumptions of each. A binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times. The binomial is a type of distribution that has two possible outcomes (the prefix “bi” means two, or twice). For example, a coin toss has only two possible outcomes: heads or tails and taking a test could have two possible outcomes: pass or fail. Excellence and Relevance Binomial Distribution Binomial distributions must also meet the following three criteria: 1. The number of observations or trials is fixed. In other words, you can only figure out the probability of something happening if you do it a certain number of times. This is common sense—if you toss a coin once, your probability of getting a tails is 50%. If you toss a coin a 20 times, your probability of getting a tails is very, very close to 100%. 2. Each observation or trial is independent. In other words, none of your trials have an effect on the probability of the next trial. 3. The probability of success (tails, heads, fail or pass) is exactly the same from one trial to another. Excellence and Relevance Binomial Distribution. Excellence and Relevance Demonstration Problem 1. A Gallup survey found that 65% of all financial consumers were very satisfied with their primary financial institution. Suppose that 25 financial consumers are sampled. If the Gallup survey result still holds true today, what is the probability that exactly 19 are very satisfied with their primary financial institution? Excellence and Relevance Demonstration Problem 2. One study by CNNMoney reported that 60% of workers have less than $25,000 in total savings and investments (excluding the value of their home). If this is true and if a random sample of 20 workers is selected, what is the probability that fewer than 10 have less than $25,000 in total savings and investments? Excellence and Relevance Mean and Standard Deviation of a Binomial Distribution A binomial distribution has an expected value or a long-run average, which is denoted by μ. The value of μ is determined by n·p. For example, if n = 10 and p =.4, then μ = n·p = (10)(.4) = 4. The long-run average or expected value means that, if n items are sampled over and over for a long time and if p is the probability of getting a success on one trial, the average number of successes per sample is expected to be n·p. If 40% of all graduate business students at a large university are women and if random samples of 10 graduate business students are selected many times, the expectation is that, on average, 4 of the 10 students would be women. Excellence and Relevance Mean and Standard Deviation of a Binomial Distribution Excellence and Relevance  Poisson Distribution Excellence and Relevance Poisson Distribution A second discrete distribution is the Poisson distribution, seemingly different from the binomial distribution but actually derived from the binomial distribution. Unlike the binomial distribution in which samples of size n are taken from a population with a proportion p, the Poisson distribution, named after Simeon-Denis Poisson (1781–1840), a French mathematician, focuses on the number of discrete occurrences over some interval or continuum. The Poisson distribution describes the occurrence of rare events. In fact, the Poisson formula has been referred to as the law of improbable events. For example, serious accidents at a chemical plant are rare, and the number per month might be described by the Poisson distribution. The Poisson distribution often is used to describe the number of random arrivals per some time interval. If the number of arrivals per interval is too frequent, the time interval can be reduced enough so that a rare number of occurrences is expected. Another example of a Poisson distribution is the number of random customer arrivals per 5-minute interval at a small boutique on weekday mornings. Excellence and Relevance Poisson Distribution The Poisson distribution has the following characteristics. It is a discrete distribution. It describes rare events. Each occurrence is independent of the other occurrences. It describes discrete occurrences over a continuum or interval. The occurrences in each interval can range from zero to infinity. The expected number of occurrences must hold constant throughout the experiment. Excellence and Relevance Poisson Distribution Examples of Poisson-type situations include the following. Number of telephone calls per minute at a small business. Number of hazardous waste sites per county in the United States. Number of arrivals at a turnpike tollbooth per minute between 3 A.M. and 4 A.M. in January on the Kansas Turnpike. Number of sewing flaws per pair of jeans during production. Number of times a tire blows on a commercial airplane per week. Each of these examples represents a rare occurrence of events for some interval. Note that although time is a more common interval for the Poisson distribution, intervals can range from a county in the United States to a pair of jeans. Some of the intervals in these examples might have zero occurrences. Moreover, the average occurrence per interval for many of these examples is probably in the single digits (1– 9). and Relevance Excellence Poisson Distribution If a Poisson-distributed phenomenon is studied over a long period of time, a long-run average can be determined. This average is denoted lambda (λ). Each Poisson problem contains a lambda value from which the probabilities of particular occurrences are determined. Although n and p are required to describe a binomial distribution, a Poisson distribution can be described by λ alone. The Poisson formula is used to compute the probability of occurrences over an interval for a given lambda value. Excellence and Relevance Poisson Distribution EXAMPLE: Suppose bank customers arrive randomly on weekday afternoons at an average of 3.2 customers every 4 minutes. What is the probability of exactly 5 customers arriving in a 4-minute interval on a weekday afternoon? The lambda for this problem is 3.2 customers per 4 minutes. The value of x is 5 customers per 4 minutes. The probability of 5 customers randomly arriving during a 4-minute interval when the long-run average has been 3.2 customers per 4-minute interval is If a bank averages 3.2 customers every 4 minutes, the probability of exactly 5 customers arriving during any one 4-minute interval is.1140. Excellence and Relevance  Poisson Distribution Using the Poisson Tables For every new and different value of lambda, there is a different Poisson distribution. However, the Poisson distribution for a particular lambda is the same regardless of the nature of the interval associated with that lambda. Because of this, over time statisticians have recorded the probabilities of various Poisson distribution problems and organized these into what we refer to as the Poisson tables. More recently, computers have been used to solve virtually any Poisson distribution problem, adding to the size and potential of such Poisson tables. Excellence and Relevance Demonstration Problem Suppose that during the noon hour in the holiday season, a UPS store averages 2.3 customers every minute and that arrivals at such a store are Poisson distributed. During such a season and time, what is the probability that more than 4 customers will arrive in a given minute? Excellence and Relevance Mean and Standard Deviation of Poisson Distribution Excellence and Relevance  Hypergeometric Distribution Excellence and Relevance Hypergeometric Distribution Another discrete statistical distribution is the hypergeometric distribution. Statisticians often use the hypergeometric distribution to complement the types of analyses that can be made by using the binomial distribution. Recall that the binomial distribution applies, in theory, only to experiments in which the trials are done with replacement (independent events). The hypergeometric distribution applies only to experiments in which the trials are done without replacement. The hypergeometric distribution, like the binomial distribution, consists of two possible outcomes: success and failure. However, the user must know the size of the population and the proportion of successes and failures in the population to apply the hypergeometric distribution. In other words, because the hypergeometric distribution is used when sampling is done without replacement, information about population makeup must be known in order to redetermine the probability of a success in each successive trial as the probability changes. Excellence and Relevance Hypergeometric Distribution The hypergeometric distribution has the following characteristics. It is a discrete distribution. Each outcome consists of either a success or a failure. Sampling is done without replacement. The population, N, is finite and known. The number of successes in the population, A, is known. Excellence and Relevance Hypergeometric Distribution In summary, the hypergeometric distribution should be used instead of the binomial distribution when the following conditions are present. 1. Sampling is being done without replacement. 2. n ≥ 5 % N. Hypergeometric probabilities are calculated under the assumption of equally likely sampling of the remaining elements of the sample space. Excellence and Relevance Demonstration Problem Suppose 18 major computer companies operate in the United States and 12 are located in California’s Silicon Valley. If 3 computer companies are selected randomly from the entire list, what is the probability that 1 or more of the selected companies are located in Silicon Valley? Excellence and Relevance  Formulas Excellence and Relevance  End of Lesson 13 Excellence and Relevance REMELYN L. ASAHID-CHENG