Читать книгу Reliability Analysis, Safety Assessment and Optimization - Enrico Zio - Страница 29

A random variable T follows the lognormal distribution if and only if the pdf (shown in Figure 1.6) of T is

Figure 1.6 The pdf of the lognormal distribution with μ=0, σ=0.954_.

(1.38)

where σ>0 is the shape parameter and μ>0 is the scale parameter of the distribution. Note that the lognormal variable is developed from the normal distribution. The random variable X=lnT is a normal random variable with parameters μ and σ. The cdf of the lognormal distribution is

(1.39)

where Φ(x) is the cdf of a standard normal random variable. If T denotes the failure time of an item with lognormal distribution, the reliability function of T will be

(1.40)

The hazard rate function is

(1.41)

The mean, μ, and variance, σ2, are

(1.42)

Example 1.3

The random variable of the time to failure of an item, T, follows the following pdf:

where t is in days and t≥0.

1 What is the probability of failure of the item in the first 100 days?

2 Find the MTTF of the item.

Solution

1 The cdf of the random variable isThe probability of failure in the first 100 days is

2 The MTTF of the item is