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When X is normally distributed with parameters and 2, find the probability density function for Y = e X. Because log Y has a normal distribution, the random variable Y is said to have a lognormal distribution with parameters and.

A probability density function (PDF) is used in probability theory to signify the random variable's likelihood of falling into a particular range of values as opposed to taking up a single value. The feature illustrates the normal distribution's probability density function and how mean and deviation are calculated.

We are given that $X~\mathcal{N}\left(渭,{蟽}^2\right)$and $Y=e^X$, Observe that $Y>0$with probability 1 . So, for any $y>0$we have that

$F_Y\left(y\right)=P(Y鈮�y)=P\left(e^X鈮�y\right)=P(X鈮�\log (y\left)\right)=F_X\left(\log \right(y\left)\right)$

where $F_X$is CDF of X. By differentiating, we have that

$f_Y\left(y\right)=\frac{d}{dy}{F}_Y\left(y\right)=f_X\left(\log \right(y\left)\right)路\frac{1}{y}=\frac{1}{蟽\sqrt{2蟺}}\exp \left(-\frac{(\log y-渭{)}^2}{2{蟽}^2}\right)路\frac{1}{y}$