Applied Statistics Using SPSS, STATISTICA, MATLAB and R

434 Appendix B - Distributions
A: The probability of obtaining a certain face is 1/6 and the occurrence of that face
at the kth Bernoulli trial obeys the geometric distribution, therefore: P (X ≥ 6)
=
− G ( 5)
= 1 − 0.6 = 0.4.
1 1/
6
B.1.4 Hypergeometric Distribution
Description: Probability of obtaining k items, of one out of two categories, in a
sample of n items extracted without replacement from a population of N items that
has D = pN items of that category (and (1−p)N = qN items from the other
category). In quality control, the category of interest is usually one of the defective
items.
Sample space: {max(0, n − N + D), …, min(n,D)}.
Probability function:
D N −D
( k )( n−k
)
N ( )
n
Np Nq ( k )( n−k
)
N ( )
hN
, D,
n ( k)
≡ P(
X = k)
= = , B. 6
k ∈{max(0, n−N+D), …, min(n,D)}.
N
From the ( n ) possible samples of size n, extracted from the population of N
items, their composition consists of k items from the interesting category and n − k
D N −D
items from the complement category. There are ( k )( n−k
) possibilities of such
compositions; therefore, one obtains the previous formula.
Distribution function:
H
N , D,
n
k
∑ hN
, D,
n
i=
max( 0,
n−
N + D)
n
( k)
=
( i)
. B. 7
Mean: np.
Variance:
⎛ N − n ⎞ N − n
npq ⎜ ⎟ , with called the finite population correction.
⎝ N −1
⎠ N −1
Example B. 4
Q: In order to study the wolf population in a certain region, 13 wolves were
captured, tagged and released. After a sufficiently long time had elapsed for the
tagged animals to mix with the untagged ones, 10 wolves were captured, 2 of
which were found to be tagged. What is the most probable number of wolves in
that region?
A: Let N be the size of the population of wolves of which D = 13 are tagged. The
number of tagged wolves in the second capture sample is distributed according to
the hypergeometric law. By studying the hN,D,n / h(N-1),D,n ratio, it is found that the
value of N that maximizes hN,D,n is:
n 10
N = D = 13 = 65.
k 2