Each individual can be characterized as a success s

 Each individual can be characterized as a success (S) or a
failure (F), and there are M successes in the population.
 A sample of n individuals is selected without replacement in
such a way that each subset of size n is equally likely to be
chosen.
 The random variable of interest is X = the number of
successes in the sample.

M N - M
x n - x
P(x) =
N
n
  
  
  
 
 
 
, for max(0, n – (N-M)) ≤ x ≤ min(n, M).
 E(X) =
M
n
N

and V(X) =
N - n M M
n 1 -
N - 1 N N
   
   
   
  
85
3.68 A certain type of digital camera comes in either a 3-
megapixel version or a 4-megapixel version. A camera store has
received a shipment of 15 of these cameras, of which 6 have 3-
megapixel resolution. Suppose that 5 of these cameras are
randomly selected to be stored behind the counter; the other 10
are placed in a storeroom. Let X = the number of 3-megapixel
cameras among the 5 selected for behind-the-counter storage.
a. What kind of distribution does X have (name and values of all
parameters)?
b. Compute P(X = 2), P(X ≤ 2), and P(X ≥ 2).
c. Calculate the mean value and standard deviation of X.
86
The Negative Binomial Distribution
 The experiment consists of a sequence of independent
trials.
 Each trial can result in either a success (S) or a failure (F).
 The probability of success is constant from trial to trial, so
P(S on trial i) = p for i = 1, 2, 3, …
 The experiment continues (trials are performed) until a total
of r successes have been observed, where r is a specified
positive integer.
 X = the number of failures that precede the r th success; the
number of successes is fixed and the number of trials is
random

r x
x + r - 1
P(x) = p q , x = 0, 1, 2,...
r - 1
 
 
 
 E(X) =
r(1 - p)
p
and V(X) =
2
r(1 - p)
p
87
3.75 Suppose that p = P(female birth) = .5. A couple wishes to
have exactly two female children in their family. They will have
children until this condition is fulfilled.