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

Answer:

Step-by-step 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,

$E(X) = e^{0.353} + \dfrac{1}{2}(0.754)^2$

$E(X) = 1.8913$

Also;

$V(X) = e^{2 \times 0.353 + 2(0.754)^2} -e^{2\times 0.353 + (0.754)^2}$

V(X) = 2.7387

∴

SD(X) = 1.6549

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

= 0.108