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.)
Best Answer
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 $[c, + \infty)$.