5' Special Discrete Distributions

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5. Special Discrete Distributions
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5.1 Bernoulli and binomial random variables

• Def X is a Bernoulli trial(or Bernoulli random
  variable) with parameter p if sample space Ss,
  f, where s is called a success and f a failure
  and X(s)1, X(f)0. The probability function of
  X is

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Bernoulli and binomial random variables

• EX 0xP(X0)1xP(X1)p
• EX2 02xP(X0)12xP(X1)p
• Var(X) EX2 (EX)2 p p2 p(1-p)

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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 p1/3.

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Bernoulli and binomial random variables

• Let X1, X2, X3, be a sequence of Bernoulli
  random variables. If, for all ji0 or 1, the
  sequence of events X1j1, X2j2, X3j3,
  are independent, we say that X1, X2, X3, and
  the corresponding Bernoulli trials are
  independent.

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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 XB(n,p) in short.
• Thm 5.1 XB(n,p)

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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
• XB(4, 8/304/15) and
• calculate P(X2)

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Bernoulli and binomial random variables

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Bernoulli and binomial random variables

• EX np
• EX(X-1) n2p2 - np2
• (similar to EX calculation)
• EX2 n2p2 - np2 np
• Var(X) EX2 (EX)2 np(1-p)

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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)

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5.2 Poisson random variable

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Poisson random variable

• Furthermore
• Def A discrete random variable X with possible
  values 0, 1, 2, 3, is called Poisson with
  parameter , gt0(XP( ) in short), if

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Poisson random variable

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Poisson random variable

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Poisson random variable

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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.

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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.

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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.

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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 EX1 so XP(1)

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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 EX1/3 so XP(1/3)

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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 EX3.9, so np3.9, n is very
  large, XP(3.9)
• (a) P(Xlt6)0.899
• (b) P(Xgt2) 0.901
• (c) P(3ltXlt6)0.646

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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
• p1/k is small
• XP(n/k)
• P(X ! 0) 1-P(X0) 1 - e-n/k

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Poisson random variable

• Poisson Process(Omitted)

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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, 0ltplt1. Then
• P(Xn)(1-p)n-1p, n1, 2, 3,
• Def The probability function p(x)(1-p)n-1p,
  n1, 2, 3, , and 0 elsewhere, is called
  geometric.
• (XGeo(p) in short)

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Other discrete random variables
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Other discrete random variables
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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 XGeo(1/13)

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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 q1-p, the probability of tails.
• (a) Find the probability that they reach a
  decision in less than n tosses.
• (b) If p1/2, what is the minimum number of
  tosses required to reach a decision with
  probability 0.95?

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Other discrete random variables

• Sol (a)The probability that they reach a
  decision on a certain round of coin tossing is
  C(3,1)pq2C(3,2)p2q3pq(pq)3pq.
• So XGeo(3pq)
• Therefore P(Xltn)1-P(X gtn)
• 1-(1-3pq)n-1
• (b)To find n such that P(Xltn)gt0.95
• or 1-P(Xgtn)gt0.95 or P(Xgtn)lt0.05
• But P(Xgtn)(1-3pq)n(1/4)n. Thus we
  have (1/4)nlt0.05. This gives n gt 2.16 hence
  the smallest n is 3.

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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, 0ltplt1. Then
• P(Xn)C(n-1, r-1)pr(1-p)n-r, nr, r1,
• Def The probability function
• p(x) C(n-1, r-1)pr(1-p)n-r, nr, r1, , is
  called negative binomial with parameters (r, p).
• (XNB(r,p) in short)
• EXr/p(see Section 9.1)

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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?

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Other discrete random variables

• Sol (a) Let X(Y) be the number of games until
  Sharon(Ann) wins 5 games. Then
• XNB(5, 0.58) and YNB(5, 0.42)
• So P(X7)P(Y7)0.170.0660.24
• (b) Let A be the event that Sharon wins and B be
  the event that the series ends in 7 games.
• P(AB)P(AB)/P(B)
• P(X7)/P(X7)P(Y7)
• 0.17/0.240.71

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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 q1-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?

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Other discrete random variables
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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, m0, 1, 2, , N?

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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
  (N1)st success occurs on the
• (N-m)(N1)(2N-m1)st trial.

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Other discrete random variables
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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 nltmin(D, N-D). Let X be the number
  of defective items drawn. Then

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Other discrete random variables

• Def Let N, D, and nnltmin(D, N-D) be positive
  integers and the probability function
• is called hypergeometric with parameters
• (N, D, n). (XHGeo(N, D, n) in short)
• EXnD/N(see Section 9.1)

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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. XHgeo(500, 25, 7)
• p(2)C(25,2)C(500-25,7-2)/C(500,7)0.04