# Double exponential function Expected value

Hi I'm having trouble calculating high moment of a double exponential function. $f(x\mid\mu,\sigma)=\frac{1}{2\sigma}e^{-\left\lvert\frac{x-\mu}{\sigma}\right\rvert}$

How do I calculate $E(X^{2009})$

I tried to calculate the moment generating function MGF but it does not work for this expectation since I have to take the derivative 2009 times!

Any suggestions? Thanks!

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$\Gamma\pars{2009 + 1} = 2009!$. The $\gamma$'s are approximated by $\gamma\pars{\alpha,x} \approx {x^{\alpha} \over \alpha}$ when $\alpha \gg 1$. Then $$\gamma\pars{2009 + 1,\pm\,{\mu \over \sigma}} \approx {\pars{\pm\,\mu/\sigma}^{2010} \over 2010}$$

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Write the integral that is used to evaluate $E(X^{2009})$ then use reduction formulae

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Not sure I understand your suggestion. –  Did Sep 4 '13 at 11:11

The generating function of $X$ is $E[\mathrm e^{tX}]=\mathrm e^{t\mu}/(1-\sigma^2t^2)$ hence $$\sum_{n\geqslant0}E[X^n]t^n/n!=\sum_{i\geqslant0}\mu^it^i/i!\cdot\sum_{j\geqslant0}\sigma^{2j}t^{2j}.$$ Equating the coefficients of $t^n$, one gets $$E[X^n]=n!\sum_{i+2j=n}\mu^i\sigma^{2j}/i!,$$ hence $$E[X^{2009}]=2009!\sum_{j=0}^{1004}\mu^{2009-2j}\frac{\sigma^{2j}}{(2009-2j)!}=2009!\sum_{k=0}^{1004}\mu^{2k+1}\frac{\sigma^{2008-2k}}{(2k+1)!}.$$ If $1004\cdot\sigma/\mu$ is large, this can be approximated by the sum of the full series, that is, $$E[X^{2009}]\approx2009!\sigma^{2009}\sinh(\mu/\sigma).$$

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Let $W=(X-\mu)/\sigma$. Then $$f_W(x) = \frac 1 2 e^{-|x|}.$$ \begin{align} E\left(X^{2009}\right) & = E\left((\sigma W+\mu)^{2009}\right) \\[10pt] & = \sum_{k=0}^{2009} \binom{2009}{k} \sigma^k E(W^k)\mu^{2009-k} \\[10pt] & = \sum_{k=0}^{2009} \binom{2009}{k} \sigma^k \mu^{2009-k} \frac 1 2 \int_{-\infty}^\infty x^k e^{-|x|} \,dx \\[10pt] & = \sum_{k=0}^{2009} \binom{2009}{k} \sigma^k \mu^{2009-k} \int_0^\infty x^k e^{-x}\,dx \\[10pt] & = \sum_{k=0}^{2009} \binom{2009}{k} \sigma^k \mu^{2009-k} k! \\[10pt] & = \sum_{k=0}^{2009} \frac{2009!}{(2009-k)!} \sigma^k\mu^{2009-k} \end{align}

I don't know how much, if anything, can be done beyond that.

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