11.2 The Binomial (Bernoulli) Distribution

The Binomial ( Bernoulli) distribution is defined as

for k = 0, 1, 2, , n and 0 < p < 1. Therefore, b( k; n; p) is non-negative.

From (11.12) and (11.11),

and thus the binomial distribution b( k; n; p) is a pdf of the discrete type.

In (11.13), we call p the probability of a success. Likewise, we call q the probability of a failure. Then,

For n = 2 and p = 1/ 2, (11.12) becomes

Moreover, for k = 0, 1 and 2, (11.15) reduces respectively to

The pdf and cdf of the binomial distribution for n = 2, p = 1/ 2, and k = 0, 1, and 2 are shown in Figures 11.1 and 11.2. We observe that they are the same as in Example 10.1.

Figure 11.1: Probability density function for the binomial distribution

Figure 11.2: Cumulative distribution function for the binomial distribution.

Solution:

Here, n = 6. Let us denote p as a success...