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.
