PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
) ( N
)
p(x) =
( M
x n − x
( ) ,
M + N
n
E(X) = n
M
M + N ,
M + N − n nMN
Var (X) =
M + N − 1 (M + N) . 2
The mgf exists, but there is no useful expression available.
1. The hypergeometric distribution is simply
# selections with x black balls
,
# possible selections
) ( N
)
=
( M
x n − x
( ) .
M + N
n
2. To see how the limits arise, observe we must have x ≤ n (i.e., no more than
sample size of black balls in the sample.) Also, x ≤ M, i.e., x ≤ min (n, M).
Similarly, we must have x ≥ 0 (i.e., cannot have < 0 black balls in sample), and
n − x ≤ N (i.e., cannot have more white balls than number in urn).
i.e. x ≥ n − N
i.e. x ≥ max (0, n − N).
3. If we sample with replacement, we would get X ∼ B ( n, p = M
M+N
)
. It is interesting
to compare moments:
finite population correction
↑
hypergeometric E(X) = np Var (X) = M+N−n [np(1 − p)]
M+N−1
binomial E(x) = np Var (X) = np(1 − p) ↓
when sample all
balls in urn Var(X) ∼ 0
4. When M, N >> n, the difference between sampling with and without replacement
should be small.
6