Todays Goals

1
Todays Goals

• Apply discrete probability distributions
• Negative binomial
• Poisson
• Homework 7 (due Wednesday March 25) Ch3 65 66
  (dont fall into the flaw of averages on (b)
  just find p( zero passengers) for (c) 68 77 1
  web problem.

2

• Daniel Bernoulli, who published the St.
  Petersburg Paradox, also discovered the Bernoulli
  Principle.
• Bernoulli Trials are named after his uncle.

3
P.M.F of the Hypergeometric Distribution

• If X is the number of Ss in a completely random
  sample of size n drawn from a population
  consisting of M Ss and (NM) Fs, then the
  probability distribution is hypergeometric
  distribution and is given by

4
Example

• In a lot of 20 units out of the production line,
  2 units are known to be defective. If the
  inspector picks a sample of 3 units at random,
  what is the distribution of the number of
  defectives in the sample?
• What is the probability that none of the chosen
  sample are defective?

5
The geometric distribution

• When you want to know the number of bernoulli
  trials until a specified event occurs for the
  first time, use the geometric distribution.
• Consider a series of bernoulli trials, with
  probability p of success.
• What is the probability that you have the first
  success on the tth trial?
• p(Tt) (1-p)t-1p

6
Example

• A radio trasmission tower is desinged for a
  50-year wind, that is a wind velocity that has
  a 1/50 chance of being exceeded.
• What is the chance that it will be exceeded for
  the first time in the 5th year?

7
Example

• A radio trasmission tower is desinged for a
  50-year wind, that is a wind velocity that has
  a 1/50 chance of being exceeded.
• What s the chance that it will be exactly 5 years
  before such velocity is exceeded (it will be
  exceeded in 5th year)?
• p(T5) (49/50)4(1/50) .018

8
Geometric Random Variable

• XNumber of failures until first success
• Y number of trials until first success (YX1)
• Recall phone call example (Y number of attempts
  until open line for international call)
• These are known as Geometric Random Variables
• The pmf of a geometric rv X with parameter
  p  P(S) is

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The Negative Binomial

• Similar to the geometric.
• It is the number of trials until the rth
  occurrence of the specified event.

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The Negative Binomial Box

• Consider the box experiment once more
• replace each cube after inspecting it, and
• sample until you got r red cubes.
• Let X equal the number of blue cubes chosen.
• If you repeated this experiment a great many
  times, the distribution of X would be negative
  binomial.

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Negative Binomial

• For this example, suppose r 2

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Negative Binomial Experiment

• A negative binomial rv and distribution are based
  on an
• experiment satisfying the following conditions
• Sample The sample size is indeterminate (varies
  from trial to trial)
• Outcomes Each trial can result in either a
  success (S) or failure (F)
• Sampling Plan
• The experiment consists of a sequence of
  independent trials
• The probability of success is constant from trial
  to trial, so P(S on trial i)  p for i  1,2,3,
• The experiment continues (trials are performed)
  until a total of r successes have been observed,
  where r is a specified positive integer.

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Negative Binomial Random Variable

• Given a negative binomial experiment, the
  negative binomial random variable X associated
  with this experiment is defined as X  the number
  of failures that precede the rth success.
• X is called a negative binomial random variable
  because, in contrast to the binomial r.v., the
  number of successes is fixed and the number of
  trials is random.

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Distribution

• The probability that is takes Xx failures to get
  r successes, with probability of success p is
• If X is a negative binomial r.v. with pmf
  nb(xr,p), then

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Example

• A partner in a consulting firm is trying to
  assemble a team of 3 analysts to come in on the
  weekend to finish a project. Each consultant has
  a probability of 0.25 of agreeing.
• What is the probability that he has to ask
  exactly 10 consultants to assemble the group of
  3?
• How many will he need to ask on average?

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Solution

• X number of rejections until 3 consultants agree
• X Negative binomial with p 0.25
• Y number of consultants asked until 3 agree
• Y X3
• PY10PX7
• EY 3EX 3 r(1-p)/p 3 3(1-0.25)/0.25
  12

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What is the probability distribution and its
parameters?

• 12 refrigerators are returned. 7 of them have a
  defective compressor. Assume that 6 are inspected
  at random. What is the probability that fewer
  than 4 of them have a defective compressor?
• Binomial
• Hypergeometric
• Geometric
• Negative Binomial

18
What is the probability distribution and its
parameters?

• 12 refrigerators are returned. 7 of them have a
  defective compressor. Assume that 6 are inspected
  at random. What is the probability that fewer
  than 4 of them have a defective compressor?
• Hypergeometric (prob of x successes out of n
  samples, no replacement)
• n6
• M 7
• N 12
• p(0,1,2,3,4) or 1-p(5,6)

19
What is the probability distribution and its
parameters?

• Suppose that only 25 of all drivers come to a
  complete stop at an intersection. Assume drivers
  arrive at the intersection randomly. What is the
  probability that at least 6 out of the next 20
  drivers will come to a complete stop?
• Binomial
• Hypergeometric
• Geometric
• Negative Binomial

20
What is the probability distribution and its
parameters?

• Suppose that only 25 of all drivers come to a
  complete stop at an intersection. Assume drivers
  arrive at the intersection randomly. What is the
  probability that at least 6 out of the next 20
  drivers will come to a complete stop?
• Binomial (prob of x successes out of n samples)
• n20
• p25
• 1-pr(0,1,2,3,4,5)

21
What is the probability distribution and its
parameters?

• A couple wants to have exactly 2 female children.
  They will continue having children until this
  condition is fulfilled. What is the expected
  value of the number of children this family will
  have?
• Binomial
• Hypergeometric
• Geometric
• Negative Binomial

22
What is the probability distribution and its
parameters?

• A couple wants to have exactly 2 female children.
  They will continue having children until this
  condition is fulfilled. What is the expected
  value of the number of children this family will
  have?
• Negative binomial (prob that there will be x
  failures before the rth success)
• r 2
• p 0.5
• EX 2