Hypergeometric Random Variables

1
Section 3.5-3.6

• Hypergeometric Random Variables
• And
• Poisson Random Variables

Note I added a few homework problems from 3.6 to
be turned in this Thursday 73, 76, 77, and
82
2
Hypergeometric Random Variables

• Let X be the number of successes obtained when
  sampling from a population of size N with M
  successes and N-M failures. X is a hypergeometric
  r.v.
• What are the possible values of X?
• What is the pmf of X?
• Note instead of p(x), pmf is labeled h(xn,M,N).

3
Mean and Variance

• The mean of a hypergeometric r.v. is
• The variance is

4
Example

• 8 women and 10 men have applied for the same job.
  The company will do 6 interviews (randomly
  chosen) per day for three days.
• What is the probability that no women are
  interviewed on the first day?
• What is the probabiity that at least 2 women are
  interviewed on the first day?
• What is the expected number and the variance of
  the number of women that will be interviewed on
  the first day?

5
Relationship between binomial and hypergeometric.

• One application of binomial is sampling from
  finite population of S and F with replacement.
• While the hypergeometric is sampling w/o
  replacement.

6
Recall HW problem

• 10000 boards, 2000 of which are green. Sample 2
  boards . Are the events 1st board is green and
  second board is green independent if sampling
  is done
• w/ replacement
• w/o replacement
• Suppose instead there are only 10 boards, 2 of
  which are green and the sampling is done
• w/replacement.
• W/o replacement

7
Compare mean and variance.

• Set pM/N. Sample size of n.
• What is the expected number of successes if
  sampling is done with replacement? Without
  replacement?
• What is the variance of the number of successes
  if sampling is done with replacement? Without
  replacement?

8
Poisson Random Variable

• Class handout.
• The Poisson r.v is a discrete r.v. with possible
  outcomes 0,1,2, and pmf
• The mean is l and the variance is l.

9
Poisson Random Variable

• The cdf is tabulated for common values of l in
  Table A.2, page 721. A piece is given here.

10
Two applications of Poisson

• The binomial (n,p) pmf and the Poisson(np) pmf
  are approximately equal when n is large and p is
  small.
• Ex. The number of three letter words in a string
  of 100 randomly generated letters.
• The number of occurrence in a fixed time
  interval. (Counting occurrences, but do not have
  distinct trials.)
• The number of calls received by a 911 operator
  during a 1 hour time period.
• In this case, l is the expected number of
  occurrence in the time interval. l is sometimes
  called the rate.

11
Example

• The number of tickets issued by a meter reader
  for parking-meter violations during a particular
  hour can be modeled by a Poisson r.v. with rate
  parameter of 4 per hour.
• What is the probability that exactly four tickets
  are given out during a particular hour?
• What is the probability that at least four
  tickets are given out during a particular hour?

12
Poisson Process

• The number of occurrences
• in a interval of length 1 is Poisson with
  parameter l.
• in an interval of length t is Poisson with
  parameter l t.
• in non-overlapping intervals are independent.
• In the previous example, suppose that the number
  of tickets issued follows a Poisson process.
• How many tickets do you expect to be given during
  a 45 minute time interval?
• What is the probability that at least three
  tickets are given out in a 45 minute time
  interval?

13
Summary Families of Discrete Random Variables

• You should know and/or be able to derive pmfs of
  Binomial and Hypergeometric random variables. You
  should know the pmf of a Poisson r.v. You should
  know when each of these probability models is
  apllicable.
• You should know mean and variance of binomial,
  hypergeometric, and Poisson r.v.s
• You should be able to use tabulated cdf for
  Binomial and Poisson random variables.
• You should know when the hypergeometric can be
  approximated by a binomial and when the binomial
  can be approximated by the Poisson
• You are not responsible for the negative binomial
  r.v. described in section 3.5