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Given a random variable $X\sim\mathcal N(0,\sigma^2)$, how can we prove that $E[X^4]=3\sigma^4$? I am having trouble even starting with the proof.

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  • $\begingroup$ Hint: $\int x^4p(x)dx$. Integrate by parts. $\endgroup$ – Yves Daoust Sep 7 '16 at 8:38
  • $\begingroup$ ... and then do partial integration a couple times :) $\endgroup$ – b00n heT Sep 7 '16 at 8:39
  • $\begingroup$ Or compute a bunch of derivatives of the moment generating function of $X$. $\endgroup$ – Surb Sep 7 '16 at 8:44
  • $\begingroup$ Hint: let $X=\sigma U$ where $U$ has standard normal distribution. Then $\mathbb EX^4=\mathbb E\sigma^4U^4=\sigma^4\mathbb EU^4$. It remains to prove that $\mathbb EU^4=3$. For this see the other hints. If in calculations parameters can be avoided then do so. $\endgroup$ – drhab Sep 7 '16 at 8:44
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    $\begingroup$ 1) Using the following web keywords "fourth order moment normal distribution proof", I have obtained at once for example this 2) The technical name for the fourth order moment is "kurtosis". $\endgroup$ – Jean Marie Sep 7 '16 at 9:07
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First with $\sigma=1$, omitting the range $(-\infty,\infty)$ for convenience and integrating twice by parts

$$E[X^4]=\frac{\displaystyle\int x^4e^{-x^2/2}dx}{\displaystyle\int e^{-x^2/2}dx}=\frac{-x^3e^{-x^2/2}+3\displaystyle\int x^2e^{-x^2/2}dx}{\displaystyle\int e^{-x^2/2}dx}=\frac{0-3xe^{-x^2/2}+3\displaystyle\int e^{-x^2/2}dx}{\displaystyle\int e^{-x^2/2}dx}=3.$$

Then by rescaling the variable,

$$3\sigma^4.$$


By observing the pattern, you easily generalize to

$$E[X^{2n}]=(2n-1)!!\sigma^{2n}.$$

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I list some hints below.

The probability density function of a normally distributed random variable with mean $0$ and variance $\sigma^2$ is

\begin{equation} f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \mathrm{e}^{-\frac{x^2}{2\sigma^2}}. \end{equation}

In general, you compute an expectation of a continuous random variable as

\begin{equation} \mathbb{E}[g(X)] = \int_{-\infty}^\infty g(x) f(x) \, \mathrm{d}x. \end{equation}

For your particular question we have that $g(x) = x^4$ and therefore

\begin{equation} \mathbb{E}[X^4] = \int_{-\infty}^\infty x^4 f(x) \, \mathrm{d}x = \frac{1}{\sqrt{2 \pi \sigma^2}} \int_{-\infty}^\infty x^4 \mathrm{e}^{-\frac{x^2}{2\sigma^2}} \, \mathrm{d}x. \end{equation}

You can solve this integral by using partial integration a number of times.

An alternative approach is to determine the moment generating function and differentiate. The moment generating function of a continuous random variable $X$ is defined as

\begin{equation} M_X(t) := \mathbb{E}[\mathrm{e}^{tX}] = \int_{-\infty}^\infty \mathrm{e}^{t x} f(x) \, \mathrm{d}x, \quad t \in \mathbb{R}. \end{equation}

For your random variable $X$ we have

\begin{equation} M_X(t) = \frac{1}{\sqrt{2 \pi \sigma^2}} \int_{-\infty}^\infty \mathrm{e}^{t x} \mathrm{e}^{-\frac{x^2}{2\sigma^2}} \, \mathrm{d}x. \end{equation}

Conveniently

\begin{equation} \mathbb{E}[X^n] = \frac{\mathrm{d}^n}{\mathrm{d}t^n} M_X(t) \bigg\vert_{t = 0}. \end{equation}

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  • $\begingroup$ When using the PDF with $g(x) = x^4$, at what point does the $\frac{1}{\sqrt{2 \pi \sigma^2}}$ term disappear? $\endgroup$ – M Smith Apr 18 '18 at 13:33

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