Prof. Vinod, Hypergeometric dist
is similar to the Binomial, has only two outcomes (sucess/Failure)
x=random variable=no of successes. There are important differences also!
1) The Trials are NOT independent for the hypergeometric.
The probability of one success in one trial denoted by p changes
from trial to trial!
Hypergeometric is a bit like pulling cards and not returning them to the deck
when the next card is drawn. This changes the probability from trial to trial.
2) Population total number of successes is known. (N is known!)
3) Hypergeometric distribution has three parameters: (N, n, A)
sample size n, population size N and
A= largest number of successes possible IN THE POPULATION
according to the description
of the random variable. The notation is k instead of A in the computer lessons.
Hypergeometric random variable
x goes from 0 to smaller of A and n =min(A,n)
min(0,3)=0, min(4,8)=4, min(8,3)=3, min(2,2)=2 etc.
EXAMPLE OF A WORD PROBLEM involving hypergeometric distribution.
There are 24 topics to study, teacher selects 10 at random
Student studies only 14. Find the probability that exactly nine of the
studied topics is on the test?
This problem is not Binomial because it is not like tossing coins.
It is more like selecting cards from a deck and not returning them back.
Once a topic to study is selected, it is not selected again. It is not Poisson
For two reasons (i) it is not a rare event like horse kick death
(ii) Also, more information is given than just the average (lambda) of something.
SOLUTION:
x=# of studied topics that are on the test.
The key phrase “studied topics on the test” defines the random variable.
N= population size = 24 number of topics out there to study
n= sample size = number selected by the teacher in a sample =10
A= largest no of successes possible in the population = 14
= those studied, cannot get more than 14 studied topics on the test!
First calculation is min(A,n)=min(14,10)=10 here.
Now we know that X random variable is in the set{0, 1, 2,... , min(A,n)=10}
That is the possible range of x is 0 to 10
P( X=9) = [(A choose x)*(N-A choose n-x)] / (N choose n)
= [2002 * 10] / 1961256
= 0.0102
Binomial is different from Poisson in following respects: True/False
1) p for Poisson is small (horsekick deaths)
2) x for Poisson goes from 0 to infinity
Hypergeometric is different from Binomial in following respects: T/F
1) Largest no of successes is n for Binomial and A for Hypergeometric
2) Range of x is different. For Hyp. Range is: 0 to the smaller of A and n
or min(A,n).