Читать книгу Reliability Analysis, Safety Assessment and Optimization - Enrico Zio - Страница 23

Consider a machine that produces a defective item with probability p (0<p<1) and produces a non-defective item with probability 1−p. Assume the events of defects in different items are mutually independent. Suppose the experiment consists of examining a sample of n of these items. Let X denote the number of defective items in the sample. Then, the random variable X follows a binomial distribution with parameters n and p and has the discrete distribution represented by the pmf in (1.14), shown in Figure 1.1. The random variable with this distribution is said to be a binomial random variable, with parameters n and p,

(1.13)

Figure 1.1 The pmf of the binomial distribution with n=5, p=0.4.

The pmf of the binomial distribution is

(1.14)

For a binomial distribution, the mean, μ, is given by

(1.15)

and the variance, σ2, is given by

(1.16)