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$Y\sim \mathcal{N}(\mu,\sigma^2)$. And $Y=\log X$

To find the probability density function of $Y$ and median of $Y$.

How I proceed: $Y=\log X$

$X=e^Y$

Using distribution function technique

$F(x)=\mathbb{P}(e^y\leq x)=\mathbb{P}(y\leq\log x)$

Now we would integrate minus infinity to $\log X$ for finding cdf and then differentiate it once we will get pdf of $X$.

But how to integrate the density and what about median??

Martín Vacas Vignolo
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demon
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    Please use MathJax to format your question, and check for spelling / typos. – owen88 Jan 02 '18 at 19:15
  • You don't need to integrate. Instead, you compute $f(x)$ by taking the derivative on both sides and using chain's rule. For more information, check log-normal distribution at https://en.wikipedia.org/wiki/Log-normal_distribution. – Math Lover Jan 02 '18 at 19:25

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It has no direct expression of x but could be expressed using Q function and the PDF is: $f_{X}(x)={1\over{\sigma\sqrt{2\pi}}x}e^{-{(lnx-\mu)^{2}}\over{2\sigma^{2}}}$

Mostafa Ayaz
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