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Let $Y$ be Log-Normal with parameters $\mu$ and $\sigma^{2}$. So $Y=e^{x}$ with $X \sim \mathcal{N}\left(\mu, \sigma^{2}\right)$. Which of the following statements is correct?

(A) The median of $Y$ is $e^{\mu}$ because the median of $X$ is $\mu$ and the exponential function is continuous and strictly increasing, so the event $Y \leq e^{\mu}$ is the same as the event $X \leq \mu .$

(B)The mode of $Y$ is $e^{\mu}$ because the mode of $X$ is $\mu$, which corresponds to $e^{\mu}$ for $Y$ since $Y=e^{x}$.

(C) The mode of $Y$ is $\mu$ because the mode of $X$ is $\mu$ and the exponential function is continuous and strictly increasing, so maximizing the PDF of $X$ is equivalent to maximizing the PDF of $Y=e^{x}$.

How do I proceed with this? I know that the median condition is not true? But do we have some expression for mode?

Reference: This problem is from Chapter 6 of Introduction to Probability by Blitzstein and Hwang.

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    $\begingroup$ Should (C) say the mode of $Y$ is $e^\mu$? Should (B) say "mean" rather than "mode"? How do you know (A) is not true? $\endgroup$
    – Henry
    Commented Aug 14, 2021 at 8:05
  • $\begingroup$ For a distribution with a continuous probability density function $f(y)$, one definition of "mode" is $\arg\max\limits_y f(y)$, the value of $y$ which maximises $f(y)$ $\endgroup$
    – Henry
    Commented Aug 14, 2021 at 8:10
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    $\begingroup$ Have you looked at the Wikipedia article for information about the mode? en.wikipedia.org/wiki/Log-normal_distribution $\endgroup$
    – user225256
    Commented Aug 14, 2021 at 13:09
  • $\begingroup$ (C) is nonsense. (B) is a subtle but (perhaps) common misunderstanding of the effects of transformations. How to show that? One way is to use Calculus: the mode will lie either at a boundary of the support, at a point where the density is not differentiable, or (as is obvious for Lognormal distributions) at a zero of the derivative of the density. That should cause you to rethink your assessment of (A). $\endgroup$
    – whuber ♦
    Commented Aug 14, 2021 at 17:43

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