A random variable Y is said to have a log-normal distribution if X = ln(Y ) has a normal

Chapter 4, Problem 4.182

(choose chapter or problem)

A random variable Y is said to have a log-normal distribution if X = ln(Y ) has a normal distribution. (The symbol ln denotes natural logarithm.) In this case Y must be nonnegative. The shape of the log-normal probability density function is similar to that of the gamma distribution, with a long tail to the right. The equation of the log-normal density function is given by f (y) = 1 y 2 e(ln(y))2/(22) , y > 0, 0, elsewhere. Because ln(y) is a monotonic function of y, P(Y y) = P[ln(Y ) ln(y)] = P[X ln(y)], where X has a normal distribution with mean and variance 2. Thus, probabilities for random variables with a log-normal distribution can be found by transforming them into probabilities that can be computed using the ordinary normal distribution. If Y has a log-normal distribution with = 4 and 2 = 1, find a P(Y 4). b P(Y > 8).