shape of the original distribution being sampled. The normal distribution is also
related to least-square estimation. It is also used as the theoretical basis for the
chi-square, student-t and F-distributions. The normal distribution is used in many
Monte Carlo simulation computer programs.
The Binomial Distribution
The binomial probability distribution is given by
b(x;n, p) = f(x) =
{(n
x
)
pxqn−x, x = 0, 1 , 2 ,.. ., n
0 , otherwise
q= 1 −p (11 .62)
It is a discrete probability distribution with parameters nand pwhere nrepresents
the number of trials and p represents the probability of success in a single trial
with q= 1 −pthe probability of failure in a single trial. For large values of nthe
binomial distribution approaches the normal distribution. In equation (11.62), the
function f(x)represents the probability of xsuccesses and n−xfailures in n-trials.
The cumulative probabilities are given by
F(x) = B(x;n, p) =
∑x
k=0
b(k;n, p), for x= 0, 1 , 2 ,... , n (11 .63)
As an exercise verify that
b(x;n, p) = b(n−x;n, 1 −p), B (x;n, p) = 1 −B(n−x−1;n, 1 −p) (11 .64)
The binomial probability law , sometimes called the Bernoulli distribution , occurs
in those application areas where one of two possible outcomes can result in a single
trial. For example, (yes, no), (success, failure), (left, right), (on, off), (defective,
nondefective), etc. For example, if there are ddefective items in a bin of N items
and an item is selected at random from the bin, then the probability of obtaining
a defective item in a single trial is p=d/N. The binomial probability distribution
involves sampling with replacement. Consequently, each time a sample of nitems
is selected from the bin containing Nitems, the probability of obtaining xdefective
items is given by equation (11.62) with p=d/N and q= 1 −d/N.
In the equation (11.62), the term
(
n
x
)
= n!
x!(n−x)!
,
(
n
0
)
= 1, and
(
0
0
)
= 1 (11 .65)
