Chapter 5 Some Discrete Probability Distributions

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Chapter 5Some Discrete Probability Distributions

• Wen-Hsiang Lu (???)
• Department of Computer Science and Information
  Engineering,
• National Cheng Kung University
• 2004/03/28

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5.1 Introduction

• Binomial distribution (Section 5.3) test the
  effectiveness of a new drug.
• Hypergeometric distribution (Section 5.4) test
  the number of defective items from a batch of
  production.
• Negative binomial distribution (Geometric
  distribution) (Section 5.5) the number of trial
  on which the first success occurs.
• Poisson distribution (Section 5.6) the number of
  outcomes occurring during a given time interval
  or in a specified region.

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5.2 Discrete Uniform Distribution

• Discrete Uniform Distribution If the random
  variable X assumes the values x1, x2, , xk, with
  equal probabilities, then the discrete uniform
  distribution is given by
• When a light bulb is selected at random from a
  box that contains a 40-watt bulb, a 60-watt bulb,
  a 75-watt bulb, and a 100-watt bulb, each element
  of the sample space S 40, 60, 75, 100 occurs
  with probability 1/4. Therefore, we have a
  uniform distribution, with
• When a die is tossed, each element of the sample
  space S 1, 2, 3, 4, 5, 6 occurs with
  probability 1/6. Therefore, we have a uniform
  distribution, with

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Discrete Uniform Distribution

• Theorem 5.1 The mean and variance of the
  discrete uniform distribution f(x k) are
• Proof
• Referring to Example 5.2 (tossing a die), we find
  that

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5.3 Binomial and Multinomial Distribution

• An experiment often consists of repeated trials,
  each with two possible outcomes that may be
  labeled success or failure.
• The most obvious application deals with the
  testing of items as they come off an assembly
  line, where each test/trial may indicate a
  defective or a nondefective item.
• The Bernoulli Process
• The experiment consists of n repeated trials.
• Each trial results in an outcome that may be
  classified as a success or a failure.
• The probability of success, denoted by p, remains
  constant from trial to trial.
• The repeated trials are independent.

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Binomial and Multinomial Distribution

• A Bernoulli trial can result in a success with
  probability p and a failure with probability q
  1 p. Then the probability distribution of the
  binomial random variable X, the number of
  successes in n independent trials, is
• Example 5.4 The probability that a certain kind
  of component will survive a given shock test is
  ¾. Find the probability that exactly 2 of the
  next 4 components tested survive.

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Binomial and Multinomial Distribution

• Binomial distribution corresponds to the binomial
  expansion of (q p)n, i.e.,
• Example 5.5 The probability that a patient
  recovers from a rare blood disease is 0.4. If 15
  people are known to have contracted this disease,
  what is the probability that (a) at least 10
  survive, (b) from 3 to 8 survive, and (c) exactly
  5 survive?

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Binomial and Multinomial Distribution

• Example 5.6 A large chain retailer purchases a
  certain kind of electronic device from a
  manufacturer. The manufacturer indicates that the
  defective rate of the device is 3
• The inspector of the retailer randomly picks 20
  items from a shipment. What is the probability
  that there will be at least one defective item
  among these 20?
• Suppose that the retailer receives 10 shipments
  in a month and the inspector randomly tests 20
  devices per shipment. What is the probability
  that there will be 3 shipments containing at
  least one defective device?
• Solution

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Binomial and Multinomial Distribution

• Theorem 5.2 The mean and variance of the
  binomial distribution b(x n, p) are
• Proof

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Binomial and Multinomial Distribution

• Example 5.7 Find the mean and variance of the
  binomial random variable of Example 5.5 (n 15,
  p 0.4), and then use Chebyshevs theorem to
  interpret the interval
• Solution

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Binomial and Multinomial Distribution

• Example 5.9 Consider the situation of Example
  5.8. The 30 are impure is merely a conjecture
  put forth by the area water board. Suppose 10
  wells are randomly selected and 6 are found to
  contain the impurity. What does this imply about
  the conjecture? Use a probability statement.
• Solution

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Binomial and Multinomial Distribution

• Multinomial Distribution If a given trial can
  result in the k outcomes E1, E2,, Ek with
  probabilities p1, p2,, pk, then the probability
  distribution of the random variables X1, X2,,
  Xk, representing the number of occurrences for
  E1, E2,, Ek in n independent trials is

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Binomial and Multinomial Distribution

• Example 5.10 The complexity of arrivals and
  departures into an airport are such that computer
  simulation is often used to model the ideal
  conditions. For a certain airport containing
  three runways it is known that in the ideal
  setting the following are the probabilities that
  the individual runways are accessed by a randomly
  arriving commercial jet Runway 1 p1 2/9,
  Runway 2 p2 1/6, Runway 3 p3 11/18. What is
  the probability that 6 randomly arriving
  airplanes are distributed in the following
  fashion? Runway 1 2 airplanes, Runway 2 1
  airplanes, Runway 3 3 airplanes.
• Solution

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5.4 Hypergeometric Distribution

• Binomial distribution the sampling with
  replacement
• Hypergeometric distribution the sampling without
  replacement
• Hypergeometric experiment1. A random sample of
  size n is selected without replacement from N
  items.2. k of the N items may be classified as
  successes and N k as failures.
• Hypergeometric random variable the number X of
  successes of a hypergeometric experiment.
• Hypergeometric distributionthe probability
  distribution of the hypergeometric variable
  X,the number of successes in a random sample of
  size n selected from N items of which k are
  labeled success and N k labeled failure.

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Hypergeometric Distribution

• Example 5.12 Lots of 40 components each are
  called unacceptable if they contain as many as 3
  defective or more. The procedure for sampling
  the lot is to select 5 components at random and
  to reject the lot if a defective is found. What
  is the probability that exactly 1 defective is
  found in the sample if there are 3 defectives in
  the entire lot?
• Solution
• Using hypergeometric distribution with x
  1, N 40, n 5, and, k 3
• So this plan is likely not desirable since
  it detects a bad lot (3 defectives) only about
  30 of the time.

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Hypergeometric Distribution

• Theorem 5.3 The mean and variance of the
  hypergeometric distribution h(x N, n, k)
  are(the proof is shown in Appendix A25)
• Exapmle 5.13 Find the mean and variance of the
  random variable of Example 5.12 (n 5, N 40,
  and k 3) and then use Chebyshevs theorem to
  interpret the interval .
• Solution

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n
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Hypergeometric Distribution

• Relationship to the Binomial Distribution
• If n is small compared to N, the nature of the N
  items changes very little in each draw. (when
  )
• The binomial distribution may be viewed as a
  large population edition of the hypergeometric
  distributions.
• Example 5.14 A manufacture of automobile tires
  reports that among a shipment of 5000 sent to a
  local distributor, 1000 are slightly blemished.
  If one purchases 10 of these tires at random from
  the distributor, what is the probability that
  exactly 3 are blemished?
• Solution

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Hypergeometric Distribution

• Multivariate Hypergeometric DistributionIf N
  items can be partitioned into the k cells A1,
  A2,, Ak with a1, a2,, ak elements,
  respectively, then the probability distribution
  of the random variable X1, X2,, Xk, representing
  the number of elements selected from A1, A2,, Ak
  in a random sample of size n, is
• Example 5.15 A group of 10 individuals are used
  for a biological case study. The group contains 3
  people with blood type O, 4 with blood type A,
  and 3 with blood type B. What is the probability
  that a random sample of 5 will contain 1 person
  with blood type O, 2 with blood type A, and 2
  with blood type B?
• Solution

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5.5 Negative Binomial and Geometric Distributions

• Negative binomial experiments the kth success
  occurs on the xth trial.
• Negative binomial random variable the number X
  of trials to produce k success in a negative
  binomial experiment.
• Negative binomial distribution If repeated
  independent trials can result in a success with
  probability p and a failure with probability q
  1 p, then the probability distribution of the
  random variable X, the number of the trial on
  which the kth success occurs, is

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Negative Binomial and Geometric Distributions

• Example 5.16 In an NBA (National Basketball
  Association) championship series, the team which
  wins four games out of seven will be the winner.
  Suppose that team A has probability 0.55 of
  winning over the team B and both teams A and B
  face each other in championship games.(a) What
  is the probability that team A will win the
  series in six games?(b) What is the probability
  that team A will win the series?(c) If both
  teams face each other in a regional playoff
  series and the winner is decided by
  winning three out of five games, what is
  the probability that team A will win a
  playoff?

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Negative Binomial and Geometric Distributions

• Geometric Distribution If repeated independent
  trials can result in a success with probability p
  and a failure with probability q 1 p, then
  the probability distribution of the random
  variable X, the number of the trial on which the
  first success occurs, is
• Example 5.17 In a certain manufacturing process
  it is known that, on the average, 1 in every 100
  items is defective. What is the probability that
  the fifth item inspected is the first defective
  item found?

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Negative Binomial and Geometric Distributions

• Example 5.18 At busy time a telephone exchange
  is very near capacity, so callers have difficulty
  placing their calls. It may be of interest to
  know the number of attempts necessary in order to
  gain a connection. Suppose that let p 0.05 be
  the probability of a connection during busy time.
  We are interested in knowing the probability that
  5 attempts are necessary for a successful call.
• Solution
• Theorem 5.4 The mean and variance of a random
  variable following the geometric distribution are
• In the system of telephone exchange, trials
  occurring prior to a success represent a cost.
• A high probability of requiring a large of number
  of attempts is not beneficial to the scientists
  or engineers.

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5.6 Poisson Distribution and the Poisson Process

• Poisson experiments Experiments yielding
  numerical values of a random variable X, the
  number of outcomes occurring during a given time
  interval or in a specified region.
• Properties of Poisson Process
• The number of outcomes in one time interval or
  specified region is independent of the number
  that occurs in any other disjoint time interval
  or region of space. In this way we say that the
  Poisson process has no memory.
• The probability that a single outcome will occur
  during a very short time interval or in a small
  region is proportional to the length of the time
  interval or the size of the region and does not
  depend on the number of outcomes occurring
  outside this time interval or region.
• The probability that more than one outcome will
  occur in such a short time interval or fall in
  such a small region is negligible.

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Poisson Distribution and the Poisson Process

• Poisson Distribution The probability
  distribution of the Poisson random variable X,
  representing the number of outcomes occurring in
  a given time interval or specified region denoted
  by t, is (mean number ? ?t)

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Poisson Distribution and the Poisson Process

• Example 5.19 During a laboratory experiment the
  average number of radioactive particles passing
  through a counter in 1 millisecond is 4. What is
  the probability that 6 particles enter the
  counter in a given millisecond? (Table A.2)
• Example 5.20 Ten is the average number of oil
  tankers arriving each day at a certain port city.
  The facilities at the port can handle at most 15
  tankers per day. What is the probability that on
  a given day tankers have to be turned away?

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Poisson Distribution and the Poisson Process

• Theorem 5.5 The mean and variance of the Poisson
  distribution
• Proof in Appendix A.26.
• Example

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The Poisson Distribution As a Limiting Form of
the Binomial

• Theorem 5.6 Let X be a binomial random variable
  with probability distribution b(x n, p).
• Proof is in Appendix A27.
• Example 5.21 In a certain industrial facility
  accidents occur infrequently. It is known that
  the probability of an accident on any given day
  is 0.005 and accidents are independent of each
  other.(a) What is the probability that in any
  given period of 400 days there will be an
  accident on one day?(b) What is the probability
  that there are at most three days with an
  accident?
• Solution

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The Poisson Distribution As a Limiting Form of
the Binomial

• Example 5.22 In a manufacturing process where
  glass products are produced, defects or bubbles
  occur, occasionally rendering the piece
  undesirable for marketing. It is known that, on
  average, 1 in every 1000 of these items produced
  has one or more bubbles. What is the probability
  that a random sample of 8000 will yield fewer
  than 7 items processing bubbles?
• Solution

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Exercise

• 9, 11, 23, 31, 43, 51, 57, 61, 69, 71.