Question

Let $Y$be a lognormal random variable (see Example 7e for its definition) and let $c>0$be a constant. Answer true or false to the following, and then give an explanation for your answer.

(a) $cY$is lognormal;

(b) $c+Y$is lognormal.

Short answer

(a) It is true that $cY$is lognormal.

(b) It is false that$c+Y$is lognormal.

Step-by-step solution

Part (a) Step 1. Given information.

$Y$isa lognormal random variable and let $c>0$is a constant.

Part (a) Step 2. Check cY is lognormal or not.

Let $Y^1=cY$

$$\begin{gathered} \log \left(Y^1\right)=\log \left(cY\right) \\ =\log c+\log Y \\ =\log c+\hspace{0.17em}X \\ \therefore Y^1=e^{X+\log c} \end{gathered}$$

Therefore,$Cy$is lognormal.

Part (b) Step 1. Check c+Y is lognormal or not.

Let $Y^1=c+Y$

$\log \left(Y^1\right)=\log \left(c+Y\right)$

Therefore, $c+Y$is not lognormal.