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# HYPERGEOMETRIC DISTRIBUTION by wgv13363

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```									   HYPERGEOMETRIC
DISTRIBUTION
PREPARED BY :A.TUĞBA GÖRE
2000432033
HYPERGEOMETRIC
DISTRIBUTION
BASİC CHARACTERİSTİCS

It models that the total number of successes
in a size sample drawn without replacement
from a finite population.
It differs from the binomial only in that the
population is finite and the sampling from the
population is without replacement.
Trials are dependent
HYPERGEOMETRIC
DISTRIBUTION

f ( X / A, B, n) 
    A      B
n x
 
x
A B
n

n= sample size
A+B=population size
A=successes in population
X=number of successes in sample
HYPERGEOMETRIC
DISTRIBUTION
Mean , Variance and Standard Deviation

n  A B A  B  n
Var ( X )             
A  B A  B 1
2

n A
  E( X ) 
A B
HYPERGEOMETRIC
DISTRIBUTION
APPROXİMATİONS
Binomial Approximation Requariments :
If A+B=N and n ≤0,05N , Binomial can be used instead of
hypergeometric distribution
Poisson Approximation Requariments:

If n  0,05N    n  20    P  0,05
Poisson can be used instead of hypergeometric distribution
HYPERGEOMETRIC
DISTRIBUTION
Example 1 :A carton contains 24 light bulbs, three of which
are defective. What is the probability that, if a sample of six is
chosen at random from the carton of bulbs, x will be defective?

P( X  x) 
        3      21
6 x
 
x
24
6

P( X  0) 
     0,40316
3        21
That is no
 
0        6
24
6                        defective
HYPERGEOMETRIC
DISTRIBUTION
(example continued)

P( X  3) 
     0,00988
3        21

 
3        3                     That is 3 will
24                         be defective.
6

Example 2:     Suppose that 7 balls are selected at random
without replacement from a box containing 5 red balls and 10 blue
balls .If X denotes the proportion of red balls in the sample,
what are the mean and the variance of X ?
HYPERGEOMETRIC
DISTRIBUTION
A=5 red       B=10 blue         A+B =15    n=7

n A B           A Bn
Var ( X )                    
A  B     2
A  B 1
7  5  10 15  7
                    0,8888
15  2
15  1
n A   75
E( X )             2,33
A  B 15
HYPERGEOMETRIC
DISTRIBUTION
Example4:Suppose that a shipment contains 5 defective items and
10 non defective items .If 7 items are selected at random without
replacement , what is the probability that at least 3 defective
items will be obtained?
N=15 (5 defective , 10 nondefective )      n=7
P( X  3)  1  P( X  2)  1  P(0)  P(1)  P(2)  0,4267

P ( 0) 
      0,0186
5        10

 
0        7
15
7

P (1) 
      0,1631
5         10

 
1         6
15
7

P ( 2) 
      0,3916
5        10

 
2        5
15
7
HYPERGEOMETRIC
DISTRIBUTION
Example 3 :If a random variable X has a hyper geometric distribution
with parameters A=8 , B=20 and n, for what value of n will Var(x) be
maximum ?
n A B         A  B  n n  8  20 8  20  n
Var ( X )                                                 
A  B   2
A  B  1 8  20  8  20  1
2

160n 28  n 
     2
        0                 n=28 or n=0 for variance
28     27                        to be maximum

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