3rd Edition: Chapter 1

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Chapter 5Special Discrete Distributions

• 5.1 Bernoulli and Binomial Random Variables
• 5.2 Poisson Random Variables
• 5.3 Other Discrete Random Variables

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5.1 Bernoulli and Binomial R.V.

• Definition
• A random variable is called Bernoulli with
  parameter p if its probability mass function is
  given by
• Mean and Variance

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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
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Binomial Random Variables
Definition If n Bernoulli trials all with
probability of success p are performed
independently, then X, the number of successes,
is called a binomial random variable with
parameters n and p. The set of possible
values of X is 0,1,2,,n, it is one of the most
important random variables.
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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 by B(n, p).
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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
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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
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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
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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 realtors claim acceptable? Sol
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Example 5.6
Suppose that jury members decide independently
and that each with probability p (0 ltp lt 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
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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
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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?
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Expectation and Variance of Binomial R.V.
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Expectation and Variance of Binomial R.V.
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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.
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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 is n
The number of misprints found by Ruby is B(n, p),
The number of misprints found by Myra is B(n, q),
The number of misprints found by both is B(n, pq),
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5.2 Poisson Random Variables
Definition A discrete random variable X with
possible values 0, 1, 2, 3, is called Poisson
with parameter ?, ? gt 0, if Properties
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Expectation and Variance of Poisson R.V.
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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

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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
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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
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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
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5.3 Other Discrete Random Variables

• Geometric R.V.

The probability mass function p(x), is called
geometric. Memoryless Property
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Property of Geometric R.V.
Geometric R.V. is the only one discrete R.V. that
has the memoryless property. Pf
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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
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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

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Negative Binomial Random Variables
The probability mass function p(x), is called
negative binomial with parameters (r, p).
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Negative Binomial R.V.
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Expectation and Variance of Negative Binomial R.V.
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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

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Example 5.21 (Attrition Ruin Problem)
Two gamblers play a game in which in each play
gambler A beat B with probability p, 0 lt p lt1,
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? Sol
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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
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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.
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Expectation of Hypergeometric R.V.
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Variance of Hypergeometric R.V.
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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
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Example 5.24
In a company of ab potential voters, a are for
the abortion and b (b lt a) are against it.
Suppose that a vote is taken to determine the
will of the majority with regard to legalizing
abortion. If n (nltb) random persons of these ab
potential voters do not vote, what is the
probability that those against abortion will
win? SolD.I.Y
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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(Xi). SolD.I.Y