Page 181 - Fundamentals of Probability and Statistics for Engineers

164 Fundamentals of Probability and Statistics for Engineers

function BINOMDIST in Microsoft 1 Excel TM 2000 gives individual binomial
probabilities given by Equation (6.2). Other statistical functions available in
Excel TM 2000 are listed in Appendix B.
The calculation of p (k) in Equation (6.2) is cumbersome as n becomes large.
X
An approximate way of determining p (k) for large n has been discussed in
X
Example 4.17 (page 106) by means of Stirling’s formula [Equation (4.78)].
Poisson approximation to the binomial distribution, to be discussed in Section
6.3.2, also facilitates probability calculations when n becomes large.
The probability distribution function (PDF), F X (x), for a binomial distribu-
tion is also widely tabulated. It is given by

m x
n
X k n k
F X x†ˆ p q ; 6:7†
k
k ˆ 0
where m is the largest integer less than or equal to x.
Other important properties of the binomial distribution have been derived in
Example 4.1 (page 77), Example 4.5 (page 81), and Example 4.14 (page 99).
Without giving details, we have, respectively, for the characteristic function,
mean, and variance,
jt
n
X t†ˆ pe ‡ q† ;
m X ˆ np; 6:8†
2
ˆ npq:
X
The fact that the mean of X is np suggests that parameter p can be estimated based
on the average value of the observed data. This procedure is used in Examples 6.2.
We mention, however, that this parameter estimation problem needs to be exam-
ined much more rigorously, and its systematic treatment will be taken up in Part B.
Let us remark here that another formulation leading to the binomial distri-
bution is to define random variable X j , j ˆ 1, 2, . . . , n, to represent the outcome
of the jth Bernoulli trial. If we let

0 if jth trial is a failure,
X j ˆ 6:9†
1 if jth trial is a success,
then the sum

X ˆ X 1 ‡ X 2 ‡ ‡ X n 6:10†

gives the number of successes in n trials. By definition, X 1 ,..., and X n are
independent random variables.

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