Question

A random variable X has a lognormal distribution if Y = ln X has a normal distribution. For this problem, assume that Y has a mean and standard deviation of 0.353 and 0.754

(a) What are the possible values X can take?
(b) What is the probability that X is between 1 and 2?

Answer 1

Explanation:

From the given information:

For X to be valid, the possible values of X should be greater than zero i.e. X > 0

If X obeys a lognormal distribution

Then; Y = In X

So,

Now,

`E(X) = e^(0.353) + (1)/(2)(0.754)^2`

`E(X) = 1.8913`

Also;

`V(X) = e^(2 * 0.353 + 2(0.754)^2) -e^(2* 0.353 + (0.754)^2)`

V(X) = 2.7387

∴

SD(X) = 1.6549

`P(1 < X < 2) = P(X < 2) - P(X < 1)`

= 0.108