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5. Special Discrete Distributions

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5. Special Discrete Distributions

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    1. 2012/5/27 1 5. Special Discrete Distributions

    2. 5.1 Bernoulli and binomial random variables Def X is a Bernoulli trial(or Bernoulli random variable) with parameter p if sample space S={s, f}, where s is called a success and f a failure and X(s)=1, X(f)=0. The probability function of X is

    3. Bernoulli and binomial random variables EX = 0xP(X=0)+1xP(X=1)=p EX2 = 02xP(X=0)+12xP(X=1)=p Var(X) = EX2 (EX)2 = p p2 = p(1-p)

    4. Bernoulli and binomial random variables Ex 5.1 If in a throw of a fair die the event of obtaining 4, 6 is called a success, and the event of obtaining 1, 2, 3, or 5 is called a failure, then is a Bernoulli random variable with parameter p=1/3.

    5. Bernoulli and binomial random variables Let X1, X2, X3, be a sequence of Bernoulli random variables. If, for all ji=0 or 1, the sequence of events {X1=j1}, {X2=j2}, {X3=j3}, are independent, we say that {X1, X2, X3, } and the corresponding Bernoulli trials are independent.

    6. Bernoulli and binomial random variables Def If n Bernoulli trials all with probability of success p are performed independently, then X, the number of successes, is called a binomial with parameters n and p. We write as X~B(n,p) in short. Thm 5.1 X~B(n,p)

    7. Bernoulli and binomial random variables Ex 5.1 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 2 of the next four customers order fish entrees? Sol: X~B(4, 8/30=4/15) and calculate P(X=2)

    8. Bernoulli and binomial random variables

    9. Bernoulli and binomial random variables EX = np E[X(X-1)] = n2p2 - np2 (similar to EX calculation) EX2 = n2p2 - np2 + np Var(X) = EX2 (EX)2 = np(1-p)

    10. 5.2 Poisson random variable Binomial probability function 1837 French mathematician Simeon-Denis Poisson introduced the following procedure to obtain the formula that approximates p(x)

    11. 5.2 Poisson random variable

    12. Poisson random variable Furthermore Def A discrete random variable X with possible values 0, 1, 2, 3, is called Poisson with parameter , >0(X~P( ) in short), if

    13. Poisson random variable

    14. Poisson random variable

    15. Poisson random variable

    16. Poisson random variable Some examples of binomial random variables that obey Poissons approximation are as follows: 1. Let X be the number of babies in a community who grow up to at least 190 centimeters. If a baby is called a success, provided that he or she grows up to the height of 190 or more centimeters, then X is a binomial random variable. Since n, the total number of babies, is large, p, the probability that a baby grows to the height of 190 centimeters or more, is small, and np, the average number of such babies, is appreciable, X is approximately a Poisson random variable.

    17. Poisson random variable 2. Let X be the number of winning tickets among the Maryland lottery tickets sold in Baltimore during one week. Then, calling winning tickets successes, we have that X is a binomial random variable. Since n, the total number of tickets sold in Baltimore, is large, p, the probability that a ticket wins, is small, and the average number of winning tickets is appreciable, X is approximately a Poisson random variable.

    18. Poisson random variable 3. Let X be the number of misprints on a document page typed by a secretary. Then X is a binomial random variable if a word is called a success, provided that it is misprinted! Since misprints are rare events, the number of words is large, and np, the average number of misprints, is of moderate values, X is approximately a Poisson random variable.

    19. Poisson random variable Ex 5.11 Every week the average number of wrong-number phone calls received by a certain mail-order house is 7. What is the probablity that they will receive (a) 2 wrong calls tomorrow; (b) at least one wrong call tomorrow? Sol: EX=1 so X~P(1)

    20. Poisson random variable Ex 5.12 Suppose that, on average, in every three pages of a book the 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: EX=1/3 so X~P(1/3)

    21. Poisson random variable Ex 5.13 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: X: the number of alpha particle during the next second Then EX=3.9, so np=3.9, n is very large, X~P(3.9) (a) P(X<=6)=0.899 (b) P(X>=2)= 0.901 (c) P(3<=X<=6)=0.646

    22. Poisson random variable Ex 5.14 Suppose that n raisins are thoroughly mixed in dough. If we bake k raisin cookies of equal size from this mixture, what is the probability that a given cookie contains at least one raisin? Sol: X: the number of raisins in the given cookie p=1/k is small X~P(n/k) P(X =! 0) = 1-P(X=0) = 1 - e-n/k

    23. Poisson random variable Poisson Process(Omitted)

    24. 5.3 Other discrete random variables Geometric Random Variable Let X be the number of experiments until the 1st success occurs and let the probability of success p, 0<p<1. Then P(X=n)=(1-p)n-1p, n=1, 2, 3, Def The probability function p(x)=(1-p)n-1p, n=1, 2, 3, , and 0 elsewhere, is called geometric. (X~Geo(p) in short)

    25. Other discrete random variables

    26. Other discrete random variables

    27. Other discrete random variables Ex 5.19 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: X~Geo(1/13)

    28. Other discrete random variables Ex 5.20 A father asks his sons to cut their backyard lawn. Since he does not specify which of the 3 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 3 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. (a) Find the probability that they reach a decision in less than n tosses. (b) If p=1/2, what is the minimum number of tosses required to reach a decision with probability 0.95?

    29. Other discrete random variables Sol: (a)The probability that they reach a decision on a certain round of coin tossing is C(3,1)pq2+C(3,2)p2q=3pq(p+q)=3pq. So X~Geo(3pq) Therefore P(X<n)=1-P(X >=n) =1-(1-3pq)n-1 (b)To find n such that P(X<=n)>=0.95 or 1-P(X>n)>=0.95 or P(X>n)<=0.05 But P(X>n)=(1-3pq)n=(1/4)n. Thus we have (1/4)n<=0.05. This gives n >= 2.16; hence the smallest n is 3.

    30. Other discrete random variables Negative Binomial Random Variable Let X be the number of experiments until the rth success occurs and let the probability of success p, 0<p<1. Then P(X=n)=C(n-1, r-1)pr(1-p)n-r, n=r, r+1, Def The probability function p(x) =C(n-1, r-1)pr(1-p)n-r, n=r, r+1, , is called negative binomial with parameters (r, p). (X~NB(r,p) in short) EX=r/p(see Section 9.1)

    31. Other discrete random variables Ex 5.21 Sharon and Ann play a series of backgammon games until one of them wins five games. Suppose that the games are independent and the probability that Sharon wins a game is 0.58. (a) Find the probability that the series ends in seven games. (b) If the series ends in seven games, what is the probability that Sharon wins?

    32. Other discrete random variables Sol: (a) Let X(Y) be the number of games until Sharon(Ann) wins 5 games. Then X~NB(5, 0.58) and Y~NB(5, 0.42) So P(X=7)+P(Y=7)=0.17+0.066=0.24 (b) Let A be the event that Sharon wins and B be the event that the series ends in 7 games. P(A|B)=P(AB)/P(B) =P(X=7)/[P(X=7)+P(Y=7)] =0.17/0.24=0.71

    33. Other discrete random variables Ex 5.22 (Attrition Ruin Problem) Two gamblers play a game in which in each play gambler A beats B with probability p and loses to B with probability q=1-p. Suppose that each play results in a forfeiture of $1 for 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?

    34. Other discrete random variables

    35. Other discrete random variables Ex 5.23 (Banach Matchbox Problem) A smoking mathematician carries 2 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 discovers that one box is empty, there are exactly m matches in the other box, m=0, 1, 2, , N?

    36. Other discrete random variables Sol: Every time that the left pocket is selected we say that a success has occurred. When the mathematician discovers that the left box is empty, the right one contains m matches iff the (N+1)st success occurs on the (N-m)+(N+1)=(2N-m+1)st trial.

    37. Other discrete random variables

    38. Other discrete random variables Hypergeometric Random Variable Suppose that, from a box containing D defective and N-D nondefective items, n are drawn at random and without replacement. Furthermore, suppose that n<=min(D, N-D). Let X be the number of defective items drawn. Then

    39. Other discrete random variables Def Let N, D, and n[n<=min(D, N-D)] be positive integers and the probability function is called hypergeometric with parameters (N, D, n). (X~HGeo(N, D, n) in short) EX=nD/N(see Section 9.1)

    40. Other discrete random variables Ex 5.24 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 2 errors? Assume that the 2nd scientist will definitely find the error of a false calculation. Sol: Let X be the number of errors found by the 2nd scientist. X~Hgeo(500, 25, 7) p(2)=C(25,2)C(500-25,7-2)/C(500,7)=0.04

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