Another useful distribution that is based on the normal is the lognormal distribution. Among other applications, this distribution is used by environmental engineers to represent the distribution of pollutant levels, by economists to represent the distribution of returns on investments, and by actuaries to represent the distribution of insurance claims.

Finding probabilities from a lognormal distribution is “as easy as falling off a log” ! If X is log normally distributed with parameters μ and σ, then Y = ln (X) is normally distributed and has mean μ and variance σ². Thus, the simplest way to work with a lognormal random variable X is to work in terms of Y = ln (X). It is easy to obtain probabilities for Y using @RISK. The expected value and variance of X are given by the following formulas:

For example, if X is log normally distributed with parameters μ = 0: 3and σ = 0.2, then Y is normal with mean 0.3 and standard deviation 0.2. Finding probabilities just means taking logs:

The mean and expected value of X are
E (X) = e0.3+0.5 (0.2)2 = 1.38
Var (X)= (e2 (0.3)) (e (0.2)2 - 1) (e (0.2 )2) = 0.077
After all that, here is a problem to work. After a hurricane, claims for property damage pour into the insurance offices. Suppose that an insurance actuary models noncommercial property damage claims (X, in dollars) as being log normally distributed with parameters μ = 10 and σ = 0.3. Claims on different properties are assumed to be independent.
a. Find the mean and standard deviation of these claims.
b. Find the probability that a claim will be greater than $50,000.
c. The company anticipates 200 claims. If the state insurance commission requires the company to have enough cash on hand to be able to satisfy all claims with probability 0.95, how much money should be in the company's reserve?

E(X) =e"+0.5 and Var(X) (e-")(er_ 1 )(e"). PN(Y > 0.336 = 0.3, = 0.2) = Pn(Z > 0.18) 0.4286