Page 103 - Applied statistics and probability for engineers

Section 3-6/Binomial Distribution 81

those that are okay. For example, the outcome OEOE indicates that the second and fourth bits are in error and the
other two bits are okay. The corresponding values for x are
Outcome x Outcome x
OOOO 0 EOOO 1
OOOE 1 EOOE 2
OOEO 1 EOEO 2
OOEE 2 EOEE 3
OEOO 1 EEOO 2
OEOE 2 EEOE 3
OEEO 2 EEEO 3
OOOO 3 EEEE 4
The event that X = 2 consists of the six outcomes:
{ EEOO, EOEO, EOOE, OEEO, OEOE, OOEE}

Using the assumption that the trials are independent, the probability of {EEOO } is
(
P EEOO) = ( ) ( ) ( ) ( 0 . 1 2 0 . 9 2 .
P E P E P O P O) = ( ) ( ) = 0 0081
Also, any one of the six mutually exclusive outcomes for which X = 2 has the same probability of occurring. Therefore,
(
P X = ) = ( 6 0 0081 ) = 0 0486.
2
.
(
−
In general, P X = x) = (number of outcomes that result in x errors) × ( . ) ( . )0 1 x 0 9 4 x .
To complete a general probability formula, only an expression for the number of outcomes that contain x errors is needed.
An outcome that contains x errors can be constructed by partitioning the four trials (letters) in the outcome into two
groups. One group is of size x and contains the errors, and the other group is of size n − x and consists of the trials that
⎛ 4⎞ 4!
are okay. The number of ways of partitioning four objects into two groups, one of which is of size x, is ⎜ ⎟ = x !( 4 x)! .
⎝
x⎠
−
Therefore, in this example,
(
P X = x) = ⎛ ⎞ 4 x⎠ 0 1 x . 0 4 − x
. ( ) ( ) 9
⎜ ⎟
⎝
⎛ 4⎞
2 2 ⎤ =
Notice that ⎜ ⎟ = 4!/ ⎡ ! ! ⎦ 6, as found above. The probability mass function of X was shown in Example 3-4 and Fig. 3-1.
⎣
2⎠
⎝
The previous example motivates the following result.
Binomial
Distribution A random experiment consists of n Bernoulli trials such that
(1) The trials are independent.
(2) Each trial results in only two possible outcomes, labeled as “success”
and “failure.”
(3) The probability of a success in each trial, denoted as p, remains constant.
The random variable X that equals the number of trials that result in a success is
a binomial random variable with parameters 0 < p < 1 and n = 1 , ,…. The prob-
2
ability mass function of X is
⎛ n⎞ n x
−
x
f x ( ) = ⎜ ⎟ p (1 − p) x = 0 , ,… , n (3-7)
1
x⎠
⎝