Hypergeometric Distribution

1
Hypergeometric Distribution

• Example
• Automobiles arrive in a dealership in lots of
  10. Five out of each 10 are inspected. For one
  lot, it is know that 2 out of 10 do not meet
  prescribed safety standards.
• What is probability that at least 1 out of the 5
  tested from that lot will be found not meeting
  safety standards?
• from Complete Business Statistics, 4th ed
  (McGraw-Hill)

2

• This example follows a hypergeometric
  distribution
• A random sample of size n is selected without
  replacement from N items.
• k of the N items may be classified as successes
  and N-k are failures.
• The probability associated with getting x
  successes in the sample (given k successes in the
  lot.)
• Where,
• k number of successes 2 n number in
  sample 5
• N the lot size 10 x number found
• 1 or 2

3
Hypergeometric Distribution

• In our example,
• _____________________________

4
Expectations of the Hypergeometric Distribution

• The mean and variance of the hypergeometric
  distribution are given by
• What are the expected number of cars that fail
  inspection in our example? What is the standard
  deviation?
• µ ___________
• s2 __________ , s __________

5
Your turn

• A worn machine tool produced defective parts for
  a period of time before the problem was
  discovered. Normal sampling of each lot of 20
  parts involves testing 6 parts and rejecting the
  lot if 2 or more are defective. If a lot from the
  worn tool contains 3 defective parts
• What is the expected number of defective parts in
  a sample of six from the lot?
• What is the expected variance?
• What is the probability that the lot will be
  rejected?

6
Binomial Approximation

• Note, if N gtgt n, then we can approximate this
  with the binomial distribution. For example
• Automobiles arrive in a dealership in lots of
  100. 5 out of each 100 are inspected. 2 /10
  (p0.2) are indeed below safety standards.
• What is probability that at least 1 out of 5
  will be found not meeting safety standards?
• Recall P(X 1) 1 P(X lt 1) 1 P(X 0)

Hypergeometric distribution Binomial distribution