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Some Discrete Probability Distributions By: Prof. Gevelyn B. Itao Probability and Statistics Discrete Uniform Distribution If the random variable X assumes the values x1, x2,…,xk, with equal probabilities, then the discrete uniform distribution is given by Probability and Statistics Discrete Uniform Distribution Example 5.1: When a light bulb is selected at random from a box that contains a 40-watt bulb, a 60-watt bulb, a 75-watt bulb, and a 100-watt bulb, each element of the sample space S = {40, 60, 75, 100} occurs with probability 1/4. Therefore, we have a uniform distribution, with Probability and Statistics Discrete Uniform Distribution Example 5.2: When a die is tossed, each element of the sample space S S = {1,2,3,4,5,6} occurs with probability 1/6. Therefore, we have a uniform distribution, with Probability and Statistics Theorem 5.1 The mean and variance of the discrete uniform distribution f (x; k) are Probability and Statistics Theorem 5.1 Example 5.3: referring to example 5.2, compute the mean and variance: Probability and Statistics Bernoulli Process An experiment that consists of repeated trials, each with two possible outcomes that may be labeled success or failure. Properties of Bernoulli Process 1. The experiment consists of n repeated trials. 2. Each trial results in 2 possible outcomes only that may be classified as a success or failure. 3. The probability of success, denoted by p, remains constant from trial to trial. 4. The repeated trials are independent. Probability and Statistics Binomial random variable The number X of successes in n Bernoulli trials Binomial distribution The probability distribution of the discrete binomial random variable. A Bernoulli trial can result in a success with probability p and a failure probability q = 1 – p. Then the probability distribution of the binomial random variable X, the number of successes in n independent trials, is Probability and Statistics Binomial distribution Example 5.4: The probability that a certain kind of component will survive a given shock test is 3/4. Find the probability that exactly 2 of the next 4 components tested survive. Probability and Statistics Binomial random variable Since p + q = 1 For P (X < r) or P (a X b), the binomial sum is provided in tables Probability and Statistics Binomial random variable Example 5.5: In In a certain city district the need for money to buy drugs is stated as the: reason for 75% of all thefts. Find the probability that among the next 5 theft cases reported in this district, a. exactly 2 resulted from the need for money to buy drugs; b. at most 3 resulted from the need for money to buy drugs. Probability and Statistics Binomial random variable Example 5.6: In testing a certain kind of truck tire over a rugged terrain, it is found that 25% of the trucks fail to complete the test run without a blowout. Of the next 15 trucks tested, find the probability that a. from 3 to 6 have blowouts; b. fewer than 4 have blowouts: c. more than 5 have blowouts. Probability and Statistics Binomial random variable Example 5.7: A nationwide survey of seniors by the University of Michigan reveals that almost 70% disapprove of daily pot smoking, according to a report in Parade. If 12 seniors are selected at random and asked their opinion, find the probability that the number who disapprove of smoking pot daily is (a) anywhere from 7 to 9: (b) at most 5; (c) not less than 8. Probability and Statistics Multinomial Distribution If a given trial can result in k outcomes E1, E2,…,Ek with probabilities p1, p2,,…pk, then the probability distribution of the random variables X1, X2, …, Xk, representing the number of occurrences for E1, E2,…,Ek in n independent trials is with Probability and Statistics Multinomial Distribution Example 5.8: According to a genetics theory, a certain cross of guinea pigs will result in red, black, and white offspring in the ratio 8:4:4. Find the probability that among 8 offspring 5 will be red, 2 black, and 1 white. Probability and Statistics Multinomial Distribution Example 5.9: The probabilities are 0.4, 0.2, 0.3, and 0.1, respectively, that a delegate to a certain convention arrived by air, bus, automobile, or train. What is the probability that among 9 delegates randomly selected at this convention, 3 arrived by air, 3 arrived by bus, 1 arrived by automobile, and 2 arrived by train? . Probability and Statistics Hypergeometric Distribution Similar to binomial distribution, except that it does not require independence among trials (i.e., it can be done without replacement) The probability distribution of the hypergeometric random variable X, in which the number of successes in a random sample of size n selected from N items of which k are labeled success and N – k labeled failure is Probability and Statistics Hypergeometric Distribution Example 5.10: To avoid detection at customs, a traveler places 6 narcotic tablets in a bottle containing 9 vitamin pills that are similar in appearance. If the customs official selects 3 of the tablets at random for analysis, what is the probability that the traveler will be arrested for illegal possession of narcotics? Probability and Statistics Hypergeometric Distribution Example 5.11: A manufacturing company uses an acceptance scheme on production items before they are shipped. The plan is a two-stage one. Boxes of 25 are readied for shipment and a sample of 3 is tested for defectives. If any defectives are found, the entire box is sent back for 100% screening. If no defectives are found, the box is shipped. a. What is the probability that a box containing 3 defectives will be shipped? b. What is the probability that a box containing only 1 defective will be sent back for screening? Probability and Statistics Hypergeometric Distribution Example 5.12: A large company has an inspection system for the batches of small compressors purchased from vendors. A batch typically contains 15 compressors. In the inspection system a random sample of 5 is selected and all are tested. Suppose there arc 2 faulty compressors in the batch of 15. (a) What is the probability that for a given sample there will be I faulty compressor? (b) What is the probability that inspection will discover both faulty compressors? Probability and Statistics Relationship between Hypergeometric and Binomial Distribution If n is very small compared to N, hypergeometric distribution approaches binomial distribution. In other words, binomial distribution is a large population version of hypergeometric distribution. The quantity k/n plays the role of the binomial parameter, p Probability and Statistics Negative Binomial Distribution Binomial Distribution Number of successes is counted for a fixed number of trials Negative Binomial Distribution The trials are repeated until a fixed number of successes occur. Probability and Statistics Negative Binomial Distribution If repeated independent trials can result in a success with probability p and a failure with probability q = 1 – p, then the probability distribution of the random variable X, the number of the trial on which the kth success occurs is Probability and Statistics Negative Binomial Distribution Example 5.13: Suppose the probability is 0.8 that any given person will believe a tale about the transgressions of a famous actress. What is the probability that a. the sixth person to hear this tale is the fourth one to believe it? b. the third person to hear this tale is the first one to believe it? Probability and Statistics Geometric Distribution If repeated independent trials can result in a success with probability p and a failure with probability q = 1 – p, then the probability distribution of the random variable X, the number of the trial on which the first success occurs is, Probability and Statistics Geometric Distribution Example 5.14: The probability that a student pilot passes the written test for a private pilot's license is 0.7. Find the probability that the student will pass the test a. on the third try; b. before the fourth try. Probability and Statistics Poisson Experiments Experiments yielding numerical values of a random variable X, the number of outcomes occurring during a given time interval or in a specified region A specified region could be a line segment, an area, a volume, or perhaps a piece of material Probability and Statistics Properties of Poisson Process 1. The number of outcomes occurring in one time interval or specified region is independent of the number that occurs in any other disjoint time interval or region of space. In other words, Poisson process has no memory. 2. The probability that a single outcome will occur during a very short time interval or in a small region is proportional to the length of the time interval or the size of the region and does not depend on the number of outcomes occurring outside this time interval or region. 3. The probability that more than one outcome will occur in such a short time interval or fall in such a small region is negligible. Probability and Statistics Poisson Distribution The probability distribution of the Poisson random variable X, representing the number of outcomes occurring in a given time interval or specified region denoted by t, is - the average number of outcomes per unit time or region Probability and Statistics Poisson Distribution Example 5.15: On average a certain intersection results in 3 traffic accidents per month. What is the probability that for any given month at this intersection a. exactly 5 accidents will occur? b. less than 3 accidents will occur? c. at least 2 accidents will occur? Probability and Statistics Poisson Distribution Example 5.16: A secretary makes 2 errors per page, on average. What is the probability that on the next page he or she will make (a) 4 or more errors? (b) no errors? Probability and Statistics Poisson Distribution and binomial distribution As n and p 0, and np remains constant, Binomial Distribution approaches a Poisson Distribution where np = t Probability and Statistics Poisson Distribution and binomial distribution Example 5.17: In a manufacturing process where glass products are produced, defects or bubbles occur, occasionally rendering the piece undesirable for marketing. It is known that, on average, 1 in every 1000 of these items produced has one or more bubbles. What is the probability that a random sample of 8000 will yield fewer than 7 items possessing bubbles? Some Continuous Probability Distributions