Download 4.1 Mathematical Expectation

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)
• 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.)
 k  N  k 
 

x  n  x 

P ( X  x )  h( x; N, n, k ) 
N 
 
n 
Where,
k = number of “successes” = 2
=5
N = the lot size = 10
n = number in sample
x = number found
= 1 or 2
Hypergeometric Distribution
• In our example,
P ( X  x )  P ( X  1)  P ( X  2)
 2 10  2   2 10  2 
 
  

1  5  1   2  5  2 

h(1;10,5,2)  h(2;10,5,2) 

10 
10 
 
 
5 
5 
= _____________________________
Expectations of the Hypergeometric
Distribution
• The mean and variance of the hypergeometric distribution
are given by
nk

N
N n
k
k
2
 
* n * (1  )
N 1
N
N
• What are the expected number of cars that fail inspection in
our example? What is the standard deviation?
μ = ___________
σ2 = __________ ,
σ = __________
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:
1. What is the expected number of defective parts in a sample of
six from the lot?
2. What is the expected variance?
3. What is the probability that the lot will be rejected?
Binomial Approximation
• Note, if N >> 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 (p=0.2) 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 < 1) = 1 – P(X = 0)
Hypergeometric distribution
Binomial distribution