Chapter 6 Some Special Discrete Distributions

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Chapter 6 Some Special Discrete Distributions

• 6.1 THE BERNOULLI DISTRIBUTION
• 6.2 THE BINOMIAL DISTRIBUTION
• 6.3 THE GEOMETRIC DISTRIBUTION
• 6.4 THE POISSON DISTRIBUTION

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6.1 THE BERNOULLI DISTRIBUTION

• 6.1.1 Bernoulli Trials
• Example 1
• The calculation of a payroll check may be correct
  or incorrect. We define the Bernoulli random
  variable for this trial so that X 0 corresponds
  to a correctly calculated check and X 1 to an
  incorrectly calculated one.
• Example 2
• A consumer either recalls the sponsor of a T.V.
  program (X 1) or does not recall (X 0)

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• Example 3
• In a process for manufacturing spoons each spoon
  may either be defective(X 1) or not (X 0).
• And, the probability distribution is

X
X

X 1 0
P(X x) p 1-p
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• Mathematically, the Bernoulli distribution is
  given by
• To calculate the mean and variance,
• Mean, ? E(X)
• Variance, ?2 E(X2) - ?
• Note Since Bernoulli distribution is determined
  by the value of p, p is the parameter of this
  distribution

P(X x) px ( 1- p)1 - x for x 1,0
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6.2 THE BINOMIAL DISTRIBUTION

• 6.2.1 The Probability Functions
• Consider an experiment which has two possible
  outcomes, one which may be termed success and
  the other failure. A binomial situation arises
  when n independent trials of the experiment are
  performed , for example
• Toss a coin 6 times
• Consider obtaining a head on a single toss as a
  success and obtaining a tail as a failure
• Throw a die 10 times
• Consider obtaining a 6 on a single throw as a
  success, and not obtaining a 6 as a failure.

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• Example
• A coin a biased so that the probability of
  obtaining a head is . The coin is tossed four
  times. Find the probability of obtaining exactly
  two heads.
• Example
• An ordinary die is thrown seven times. Find the
  probability of obtaining exactly three sixes.

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• Example
• The probability that a marksman hits a target is
  p and the probability that he misses is q, where
  q 1 p. Write an expression for the
  probability that, in 10 shots, he hits the target
  6 times.

If the probability that an experiment results in
a successful outcome is p and the probability
that the outcome is a failure is q, where q 1
p, and if X is the random variable the number of
successful outcomes in n independent trials,
then the probability function of X is given by
P(X x) for x 0,1,2,,n
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• Example
• If p is the probability of success and q 1- p
  is the probability of failure, find the
  probability of 0,1,2,,5 successes in 5
  independent trials of the experiment. Comment
  your answer.

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• If X is distribution in this way, we write
• n and p are called the parameters of the
  distribution.
• Sometimes, we will use b(x n,p) to represent the
  probability function when X?Bin(n,p). i.e. b(x
  n,p) P(X x)

X ? Bin(n,p) where n is the number of
independent trials and p is the probability of a
successful outcome in one trial
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• Example
• The probability that a person supports Party A is
  0.6. Find the probability that in a randomly
  selected sample of 8 voters there are
• (a) exactly 3 who support Party A,
• (b) more than 5 who support Party A.
• Example
• A box contains a large number of red and yellow
  tulip bulbs in the ratio 13. Bulbs are picked at
  random from the box. How many bulbs must be
  picked so that the probability that there is at
  least one red tulip bulb among them is greater
  than 0.95?

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• 6.2.2 Mean and Variance of the Binomial
  Distribution
• Proved it by yourself. )

? np ?2 npq
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• Example
• If the probability that it is find day is 0.4,
  find the expected number of find days in a week,
  and the standard deviation.
• Example
• The random variable X is such that X ? Bin(n,p)
  and E(X) 2, Var(X) . Find the values of n and
  p, and P(X 2).

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• 6.2.3 Application

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• C.W. Applications of Binomial distributions
• Throughout this unit, daily lift examples and
  discussions are the essential features.
• Binomial Distribution
• Q1 The probability that a salesperson will sell a
  magzine subscription to someone who has been
  randomly selected from the telephone directory is
  0.1. If the salesperson calls 6 individuals this
  evening, what is the probability that
• (I) There will be no subscriptions will be
  sold?
• (II) Exactly 3 subscriptions will be sold?
• (III) At least 3 subscriptions will be sold?
• (IV) At most 3 subscriptions will be sold?

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6.3 THE GEOMETRIC DISTRIBUTION

• 6.3.1 The Probability Function
• A geometric distribution arises when we have a
  sequence of independent trials, each with a
  definite probability p of success and probability
  q of failure, where q 1 p. Let X be the
  random variable the number of trials up to and
  including the first success.
• Now,
• P(X 1) P(success on the first trial) p
• P(X 2) P(failure on first trial, success on
  second) q p
• P(X 3) ___________________________
• P(X 4) __________________________
• .
• .
• P(X x ) __________________________

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• p is the parameter of the distribution.
• If X is defined in this way, we write

P(X x) qx 1 p, x 1,2,3, where q 1 p.
X ? Geo(p)
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• 6.3.2 Mean and Variance
• Proved it by yourself. ?

? and ?2
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• Example
• The probability that a marksman hits the bulls
  eye is 0.4 for each shot, and each shot is
  independent of all others. Find
• the probability that he hits the bulls eye for
  the first time on his fourth attempt,
• the mean number of throws needed to hit the
  bulls eye, and the standard deviation,
• the most common number of throws until he hits
  the bulls eye.

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• Example
• A coin is biased so that the probability of
  obtaining a head is 0.6. If X is the random
  variable the number of tosses up to and
  including the first head, find
• P(X ? 4),
• P(X gt 5),
• The probability that more than 8 tosses will be
  required to obtain a head, given the more than 5
  tossed are required.

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• Example
• In a particular board game a player can get out
  of jail only by obtaining two heads when she
  tosses two coins.
• Find the probability that more than 6 attempts
  are needed to get out of jail.
• What is the smallest value of n if there is to be
  at least a 90 chance of getting out of jail on
  or before the n th attempt.

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• C.W. Application of Geometric Distribution
• 1)The probability that a student will pass a test
  on any trial is 0.6. What is the probability that
  he will eventually pass the test on the second
  trial?
• 2)Suppose the probability that Hong Kong
  Observatory will make correct daily whether
  forecasts is 0.8. In the coming days, what is the
  probability that it will make the first correct
  forecast on the fourth day?

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6.4 THE POISSON DISTRIBUTION

• see textbook
• Example
• Verify that if X?Po(?), then X is a random
  variable.
• Example
• If X?Po(?) find (a) E(X), (b) E(X2), (c) Var(X).
• From above example , we can conclude that the
  MEAN and VARIANCE of the Poisson distribution are
  ? and ? respectively.

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C.W. Application of Poisson Distribution

• 1)The average number of claims per day made to
  the Insurance Company for damage or losses is
  3.1. What is the probability that in any given
  day
• fewer than 2 claims will be made?
• exactly 2 claims will be made?
• 2 or more claims will be made?
• more than 2 claims will be made?

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• 2) Based on past experience, 1 of the
  telephone bills mailed to house-holds in Hong
  Kong are incorrect. If a sample of 10 bills is
  selected, find the probability that at least one
  bill will be incorrect. Do this using two
  probability distributions (the binomial and the
  Poisson) and briefly compare and explain your
  result.