Lognormal distribution

Consider a random variable X having a lognormal distribution with a mean H-x and standard deviation ox, that is, X ~ LN(^x, ox). For a lognormal random variable X, its logarithmic transform Y = ln(X) leads to a normal distribution for Y. The PDF of X is given in Eq. (2.65). In the log-transformed space, the mean and standard deviation of ln(X) can be computed, in terms of цx and ox, by Eqs. (2.67a) and (2.67b). Since Y = ln(X) is normally distributed, the generation of lognormal random variates from X ~ LN(^x, ox) can be obtained by the following steps:

1. Calculate the mean ^ln x and standard deviation oin x of log-transformed vari­able ln(X) by Eqs. (2.67a) and (2.67b), respectively.

2. Generate the standard normal variate г from N(0,1).

3. Compute y = ,u. lnx + olnxZ.

4. Compute the lognormal random variate x = ey.

6.1.3 Exponential distribution

The exponential distribution is used frequently in reliability computation in the framework of time-to-failure analysis. It is often used to describe the stochastic behavior of time to failure and time-to-repair of a system or component. A ran­dom variable X having an exponential distribution with parameter в, denoted by X ~ EXP(e), is described by Eq. (2.79). By the CDF-inverse method,

u = Fx (x) = 1 – e-x/e (6.18)

so that

X = – в ln( 1 – U)

Since 1 – U is distributed in the same way as U, Eq. (6.19) is reduced to

X = – в ln(U) (6.20)

Equation (6.20) is also valid for random variables with the standard exponential distribution, that is, V ~ exp(e = 1). The algorithm for generating exponential variates is

1. Generate uniform random variate u from U(0,1).

2. Compute the standard exponential random variate v = – ln(u).

3. Calculate x = vft.