Chapter 5 Special Discrete Distributions

1. Chapter 5Special Discrete Distributions 5.1 Bernoulli and Binomial Random Variables 5.2 Poisson Random Variables 5.3 Other Discrete Random Variables Discrete R.V.

2. 5.1 Bernoulli and Binomial R.V. Definition A random variable is called Bernoulli with parameter pif its probability mass function is given by MeanandVariance： Discrete R.V.

3. Example 5.1 If in a throw of a fair die the event of obtaining 4 or 6 is called a success, and the event of obtaining 1, 2, 3, or 5 is called a failure, then is a Bernoulli R.V. with parameter = ? What is its PMF? Mean? Variance? Sol： Discrete R.V.

4. Binomial Random Variables Definition If nBernoulli trials all with probability of success p are performed independently, then X, the number of successes, is called a binomial random variable with parametersnandp. The set of possible values of X is {0,1,2,…,n}, it is one of the most important random variables. Discrete R.V.

5. Theorem 5.1 Let X be a binomial random variable with parameters n and p. Then p(x), the probability mass function of X, is Definition The function p(x) given by the above is called the binomial probability mass function with parameters (n, p). Denoted byB(n, p). Discrete R.V.

6. Example 5.2 A restaurant serves 8 entrees of fish, 12 of beef, and 10 of poultry. If customers select from these entrees randomly, what is the probability that two of the next four customers order fish entrees? Sol： Discrete R.V.

7. Example 5.3 In a country hospital 10 babies, of whom six were boys, were born last Thursday. What is the probability that the first six births were all boys? Assume that the events that a child born is a girl or is a boy are equiprobable. Sol： Discrete R.V.

8. Example 5.4 In a small town, out of 12 accidents that occurred in June 1986, four happened on Friday the 13th. Is this a good reason for a superstitious person to argue that Friday the 13th is inauspicious? Sol： Discrete R.V.

9. Example 5.5 A realtor claims that only 30% of the houses in a certain neighborhood are appraised at less than $200,000. A random sample of 20 houses from that neighborhood is selected and appraised. The results in (thousands of dollars) are as follows: 285 156 202 306 276 562 415 245 185 143 186 377 225 192 510 222 264 198 168 363 Based on these data, is the realtor’s claim acceptable? Sol： Discrete R.V.

10. Example 5.6 Suppose that jury members decide independently and that each with probability p (0 <p < 1) makes the correct decision. If the decision of the majority is final, which is preferable: a three-person jury or a single juror? Sol： Discrete R.V.

11. Example 5.7 Let p be the probability that a randomly chosen person is against abortion, and let X be the number of persons against abortion in a random sample of size n. Suppose that, in a particular random sample of n persons, k are against abortion. Show that P(X = k) is maximum for p = k/n. Pf： Discrete R.V.

12. Expectation and Variance of Binomial R.V. If X is a binomial random variable, with parameters (n, p), then what are the mean and variance? Discrete R.V.

13. Expectation and Variance of Binomial R.V. Discrete R.V.

14. Expectation and Variance of Binomial R.V. Discrete R.V.

15. Example 5.8 A town of 100,000 inhabitants is exposed to a contagious disease. If the probability that a person becomes infected is 0.04, what is the expected number of people who become infected? Sol： What distribution of no. of people infected? Ans:E(X)= np =4000. Discrete R.V.

16. Example 5.9 Two proofreaders, Ruby and Myra, read a book independently and found r and m misprints, respectively. Suppose that the probability that a misprint is noticed by Ruby is p and the probability that is noticed by Myra is q. where these two probabilities are independent. If the number of misprints noticed by both Ruby and Myra is b, estimate the number of unnoticed misprints. Sol： Suppose the total number of misprints isn The number of misprints found by Ruby isB(n, p), The number of misprints found by Myra isB(n, q), The number of misprints found by both isB(n, pq), Discrete R.V.

17. 5.2 Poisson Random Variables Definition A discrete random variable X with possible values 0, 1, 2, 3, … is called Poisson with parameter , > 0, if Properties Discrete R.V.

18. Expectation and Variance of Poisson R.V. Discrete R.V.

19. Example 5.10 Every week the average number of wrong-number phone calls received by a certain mail-order house is 7. What is the probability that they will receive two wrong calls tomorrow; (b) at least one wrong call tomorrow? Sol： Discrete R.V.

20. Example 5.11 Suppose that, on average, in every 3 pages of a book there is one typographical error. If the number of typographical errors on a single page of the book is a Poisson random variable, what is the probability of at least one error on a specific page of the book? Sol： Discrete R.V.

21. Example 5.12 The atoms of a radioactive element are randomly disintegrating. If every gram of this element, on average, emits 3.9 alpha particles per second, what is the probability that during the next second the number of alpha particles emitted from 1 gram is (a) at most 6; (b) at least 2; (c) at least 3 and at most 6 ? Sol： Discrete R.V.

22. Example 5.13 Suppose that n raisins are thoroughly mixed in dough. If we bake k raisin cookies of equal sizes from this mixture, what is the probability that a given cookie contains at least one raisin? Sol： Discrete R.V.

23. 5.3 Other Discrete Random Variables Geometric R.V. The probability mass function p(x), is called geometric. Memoryless Property Discrete R.V.

24. Property of Geometric R.V. Geometric R.V. is the only one discrete R.V. that has the memoryless property. Pf： Discrete R.V.

25. Example 5.18 From an ordinary deck of 52 cards we draw cards at random, with replacement, and successively until an ace is drawn. What is the probability that at least 10 draws are needed? Sol： Discrete R.V.

26. Example 5.19 • A father asks his sons to cut their backyard lawn. Since he • does not specify which of the three sons is to do the job, • each boy tosses a coin to determine the odd person, who • must then cut the lawn. In the case that all three get • heads or tails, they continue tossing until they reach a • decision. Let p be the probability of heads and q = 1p, the • probability of tails. • Find the probability that they reach a decision in less than n tosses. • If p = 1/2, what is the minimum number of tosses required to reach a decision with probability 0.95? • Sol： Discrete R.V.

27. Negative Binomial Random Variables The probability mass function p(x), is called negative binomial with parameters (r, p). Discrete R.V.

28. Negative Binomial R.V. Discrete R.V.

29. Expectation and Variance of Negative Binomial R.V. Discrete R.V.

30. Example 5.20 • Sharon and Ann play a series of backgammon games until • one of them win 5 games. Suppose that the games are • independent and the probability that Sharon wins a game • is 0.58. • Find the probability that the series ends in 7 games. • If the series ends in 7 games, what is the probability that Sharon wins? • Sol： Discrete R.V.

31. Example 5.21 (Attrition Ruin Problem) Two gamblers play a game in which in each play gambler A beat B with probabilityp, 0 < p <1, and loses to B with probabilityq = 1p. Suppose that each play results in a forfeiture of $1for the loser and in no change for the winner. If player A initially has a dollars and player B has b dollars, what is the probability that B will be ruined? Sol： Discrete R.V.

32. Example 5.22 A smoking mathematician carries two matchboxes, one in his right pocket and one in his left pocket. Whenever he wants to smoke, he selects a pocket at random and takes a match from the box in that pocket. If each matchbox initially contains N matches, what is the probability that when the mathematician for the first time discover that one box is empty, there exactly m matches in the other box, m =0,1,2,…,N. Sol： D.I.Y Discrete R.V.

33. Hypergeometric Random Variables Let N, D, and n be positive integers with n min(D, ND). Then is said to be a hypergeometric probability mass function. Discrete R.V.

34. Expectation of Hypergeometric R.V. Discrete R.V.

35. Variance of Hypergeometric R.V. Discrete R.V.

36. Example 5.23 In 500 independent calculations a scientist has made 25 errors. If a second scientist checks 7 of these calculations randomly, what is the probability that he detects two errors? Assume that the second scientist will definitely find the error of a false calculation. Sol： Discrete R.V.

37. Example 5.24 In a company of a+b potential voters, a are for the abortion and b (b < a) are against it. Suppose that a vote is taken to determine the will of the majority with regard to legalizing abortion. If n (n<b) random persons of these a+b potential voters do not vote, what is the probability that those against abortion will win? Sol：D.I.Y Discrete R.V.

38. Example 5.25 Professors Davidson and Johnson from the University of Victoria in Canada gave the following problem to their students in a finite math course: An urn contains N balls of which B are black and NB are white; n balls are chosen at random and without replacement from the urn. If X is the number of black balls chosen, find P(X=i). Sol：D.I.Y Discrete R.V.