64 2. FUNDAMENTAL RELIABILITY MATHEMATICS
named as “failure” and represented by a numerical number “0” with a probability q. Since this
random variable has only two outcomes, we have:
p C q D 1: (2.53)
e repeated independent n trials of such a random variable are called n-Bernoulli trials. e
sample space of one such random variable has two sample points “1” as “success” and “0” as
“failure.” en, the sample space of n-Bernoulli trials will have 2
n
sampling points. For exam-
ple, 3-Bernoulli trials will have 2
3
D 8 sample points, which includes 000, 001, 010, 011, 100,
101, 110, and 111. Let X represents the number of successes in the n-Bernoulli trials. en, the
discrete random variable X of the n-Bernoulli trials can be described by the Binomial distribu-
tion.
Binomial distribution: For n-Bernoulli trials where a “success” is represented by a numerical
number “1” with a probability p, the PMF of the random variable X in the n-Bernoulli trials
with the number of “success” x follows the Binomial distribution expressed as:
p
.
X D x
/
D
n
x
!
p
x
.1 p/
nx
for x D 0; 1; 2; : : : ; n; (2.54)
where X is a discrete random variable which is equal to the number of “success” in the n-
Bernoulli trials. x is the realizing value of the random variable X.
n
x
is the number of possible
combinations of x objects from a set of n objects and is equal to
nŠ
xŠ.n x/Š
. p is the probability
of a “success” in one trial.
e CDF of the Binomial distribution per Equation (2.43) will be:
F
.
x
/
D P
.
X x
/
D
kDx
X
kD0
"
n
k
!
p
knk
#
: (2.55)
e Binomial distribution is fully defined by two distribution parameters n and p. e mean
X
and standard deviation
X
of a Binominal distribution X are determined by the following two
equations:
X
D E
.
X
/
D np (2.56)
X
D
p
var
.
X
/
D
p
np.1 p/: (2.57)
In Microsoft Excel, the functions for calculating PMF and the CDF of a Binomial distribution
are:
p
.
x
/
D BINOM:DIST
.
x; n; p; FALSE
/
(2.58)
F
.
x
/
D BINOM:DIST
.
x; n; p; TRUE
/
: (2.59)