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Chapter 7: Q. 7.49 (page 363)

The positive random variable X is said to be a lognormal random variable with parameters渭 and蟽2 iflog(X) is a normal random variable with mean 渭and variance role="math" localid="1647407606488" 蟽2. Use the normal moment generating function to find the mean and variance of a lognormal random variable

Short Answer

Expert verified

The mean and variance of a lognormal random variable value are E(X)=emu+蟽2/2and

Variancerole="math" localid="1647408572874" (X)=e2渭+蟽2e蟽2-1.

Step by step solution

01

Given Information

Use the normal moment generating function to find the mean and variance of a lognormal random variable.

02

Explanation

Define, Y=logX

Since we know thatXis log-normal random variable with parameters 渭and 蟽2, we have that Y~N渭,蟽2.

Observe the following equality. For t鈮�0

We have that,

EXt=EeYt=EetY=MY(t)

=exp渭t+蟽2t22.

03

Explanation

Now we have that,

E(X)=exp渭+蟽22and

EX2=exp2渭+2蟽2

Which implies, variancerole="math" localid="1647408375853" (X)=EX2-E(X)2

=exp2渭+2蟽2-exp2渭+蟽2

鈬�Var(X)=e2渭+蟽2e蟽2-1.

04

Final answer

The mean and variance of a lognormal random variable value areE(X)=emu+蟽2/2and

variance(X)=e2渭+蟽2e蟽2-1.

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