Chapter 5.4
Binomial Distribution
A special application of a discrete probability distribution is the
binomial distribution.
To start, introduce the random variable
that takes two outcomes:
B
X
x = 1
“success”
x = 0
“failure”
The probability distribution function of
is:
B
X
for 0 <
p
< 1
(the probability of success)
p
)
X
(
P
B
=
=
1
p
1
)
X
(
P
B
−
=
=
0
This is known as the Bernoulli distribution.
The mean and variance are calculated as:
p
p
p
)
x
(
P
x
)
X
(
E
x
B
X
B
=
−
⋅
+
⋅
=
=
=
μ
∑
)
(1
0
1
)
p
1
(
p
p
p
p
p
p
p
)
x
(
P
x
)
X
(
E
)
X
(
Var
2
2
2
x
2
2
X
2
B
B
B
−
=
−
=
−
−
⋅
+
⋅
=
−
=
μ
−
=
∑
)
(1
(0)(0)
(1)(1)
Chapter 5
17

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Now consider that a random experiment with the outcome of success
or failure is repeated
n
times.
Each trial produces success or failure with probabilities
p
and (1
−
p
)
respectively. Assume independence so that the result of one trial
does not influence the result of any other trial.
Let the random variable
X
be the number of successes in
n
trials.
The probability distribution function of
X
is defined as:
)
(
P
)
x
(
P
trials
t
independen
n
in
successes
x
=
for
x
= 0, 1, 2, . . . ,
n
This is known as the
binomial distribution
.
Chapter 5
18