is the cumulative distribution function for 
, then
:
th moment of ![E[Y^n]=\displaystyle\int_{-\infty}^\infty y^nf_Y(y)\,\mathrm dy=\frac1{\sqrt{2\pi}}\int_0^\infty y^{n-1}e^{-\frac12(\ln y)^2}\,\mathrm dy](/tpl/images/0407/6099/b214b.png)
, so that 
and 
:![E[Y^n]=\displaystyle\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty e^{nu}e^{-\frac12u^2}\,\mathrm du=\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty e^{nu-\frac12u^2}\,\mathrm du](/tpl/images/0407/6099/9795b.png)
![E[Y^n]=\displaystyle\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty e^{\frac12(n^2-(u-n)^2)}\,\mathrm du=\frac{e^{\frac12n^2}}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-\frac12(u-n)^2}\,\mathrm du](/tpl/images/0407/6099/b55b5.png)
is exactly the PDF of a normal distribution with mean 
:![E[U^0]=\displaystyle\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-\frac12(u-n)^2}\,\mathrm du=1](/tpl/images/0407/6099/40eb9.png)
![E[Y^n]=e^{\frac12n^2}](/tpl/images/0407/6099/03ef6.png)
Let x ~ n(0, 1) and y = ex. y is called a log-normal random variable.
(a) find the probability...
Mathematics, 07.12.2019 04:31 P05182010
Let x ~ n(0, 1) and y = ex. y is called a log-normal random variable.
(a) find the probability density function of y .
(b) find the nth moment e(yn) of y .
hint. do not compute the moment generating function of y . instead relate
the nth moment of y to an expectation of x that you know.
Answers: 1
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