Math Help - [SOLVED] 4th central moment

1. [SOLVED] 4th central moment

Hi,

I need to find $\mu_4$ for an iid sample from an Exp (1) distribution. I am given the hint to "think" Gamma functions. I would appreciate any directional advice! (I know it's the 4th central moment, but not sure how to get toward Gamma functions...)

Thanks!

2. Originally Posted by Statistik
Hi,

I need to find $\mu_4$ for an iid sample from an Exp (1) distribution. I am given the hint to "think" Gamma functions. I would appreciate any directional advice! (I know it's the 4th central moment, but not sure how to get toward Gamma functions...)

Thanks!
The forth central moment is given by

$\mathbb{E}[X^4]=\int_{0}^{\infty}x^4f(x)dx$

The density function for an exponential is

$f(x)=\lambda e^{-\lambda x}$

So putting the two facts together gives

$\mathbb{E}[X^4]=\lambda\int_{0}^{\infty}x^4e^{-\lambda x}dx$

From here make the $u=-\lambda x \implies du =-\lambda dx$

This should look alot like a gamma function.

3. You can also obtain the fourth moment of a gamma by differentiating the MGF four times.

$M(t)=(1-\beta t)^{-\alpha}$