Fundamentals of Probability and Statistics for Engineers

a sequence of Bernoulli trials can be symbolically represented by

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

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 D istribution

The probability distribution of a random variable X representing the number of
successes in a sequen ce 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 pX (k), the probability of having exactly k successes in n trials.
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
second S,..., and, finally, n k 1 choices for the position of the kth S. The
total number of possible arrangements is thus n(n 1)... (n k 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 k!. Hence, the number of ways in
which k successes can happen in n trials is

and the probability associated with each is H ence, we have

162 FundamentalsofProbabilityandStatisticsforEngineers

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