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Let $Y$ be a lognormal random variable and let $c>0$ be a constant. Is $c+Y$ lognormal?

My attempt:

We need to check if $X=\log(c+Y)$ is a normal random variable. Let $F_X$ denote the cumulative distribution function of $X$.

Since $Y$ is lognormal, $\log (Y)$ is a normal random variable. In particular this means that $Y>0$.

$\begin{aligned}[t] F_X(x)&=P\{X\le x\} \\ &=P\{\log(c+Y)\le x\} \\ &=P\{c+Y\le e^x\} \\ &=P\{Y\le e^x-c\} \end{aligned}$

We have already noted that $Y>0$. So, if $e^x-c$ is negative, $P\{Y\le e^x-c\}=0$.

That is, $F_X(x)=0$ if $x<\log c$.

But the cumulative distribution function of a normal random variable is always positive. So, $X$ is not a normal random variable. So, $c+Y$ is not lognormal.

Is this correct? (I'm very new to probability.)

Siddhartha
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1 Answers1

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The support of a lognormal distribution must be $[0, + \infty)$. If $Y$ is lognormal, then $Y >0$; so $Y+c > c >0$ and $Y+c$ cannot be lognormal because it would have support $[c, + \infty)$.

mlc
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