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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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