What's wrong with my calculation? $E[X^3]$ where $Z\sim N(0,1)$

by superuser123   Last Updated February 11, 2019 10:20 AM

Let $X\sim N(0,1)$, compute $E[X^3]$:

My attempt:

$$ E[X^3]=\int_{-\infty}^{\infty}x^3\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}dx \\ x^2=t \Rightarrow dx=\frac{dt}{2x}\\ E[X^3]=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty} t\frac{1}{2}e^{-\frac{t}{2}}dt=\frac{1}{\sqrt{2\pi}}E[Y] \\ Y\sim exp(\frac{1}{2}) \\ \Rightarrow E[X^3]=\frac{1}{\sqrt{2\pi}}\frac{1!}{0.5^1}=\frac{2}{\sqrt{2\pi}} $$

I used the fact that if $X\sim exp(\lambda)$ then $E[X^n]=\frac{n!}{\lambda^n}$

But the actual is result is 0! why is that? what am I missing here??


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