Page 179 - Fundamentals of Probability and Statistics for Engineers

162 Fundamentals of Probability and Statistics for Engineers

a sequence of Bernoulli trials can be symbolically represented by

SSFFSFSSS FF
FSFSSFFFS SF
.
.
.
and, owing to independence, the probabilities of these possible outcomes are
easily computed. For example,

P SSFFSF FF†ˆ P S†P S†P F†P F†P S†P F† P F†P F†
ˆ ppqqpq qq:

A number of these possible outcomes with their associated probabilities are
of practical interest. We introduce three important distributions in this connection.

6.1.1 BINOMIAL DISTRIBUTION

The probability distribution of a random variable X representing the number of
successes in a sequence of n Bernoulli trials, regardless of the order in which they
occur, is frequently of considerable interest. It is clear that X is a discrete random
variable, assuming values 0, 1, 2, .. . , n. In order to determine its probability mass
function, consider p (k), the probability of having exactly k successes in n trials.
X
This event can occur in as many ways as k letters S can be placed in n boxes.
Now, we have n choices for the position of the first S, n 1 choices for the
k
second S, .. . , and, finally, n ‡ 1 choices for the position of the kth S. The
k
total number of possible arrangements is thus n(n 1) . . . (n ‡ 1). However,
as no distinction is made of the Ss that are in the occupied positions, we must
divide the number obtained above by the number of ways in which k Ss can be
arranged in k boxes, that is, k(k 1) .. . 1 ˆ k!. Hence, the number of ways in
which k successes can happen in n trials is

n n 1† n k ‡ 1† n!
ˆ ; 6:1†
k! k! n k†!

k n k
and the probability associated with each is p q . Hence, we have

n k n k

p k†ˆ p q ; k ˆ 0; 1; 2; ... ; n; 6:2†
X
k

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