# Thread: determine distribution function and nth moment

1. ## determine distribution function and nth moment

Hi, I'm so struggling with this quetsion, can anyone please kindly help me? Thanks a lot.

Let $\alpha, \beta >0$ and
$f(x)=\left\{\begin{array}{cc}\frac{\alpha}{\beta}( \frac{x}{\alpha})^{\alpha -1} \exp(-(\frac{x}{\beta})^\alpha),&\mbox{ if } x>0\\0,&\mbox{otherwise}\end{array}\right$
and suppose the random variable X has density f.

(a)Determine the distribution function $F(x)$ of X
(b)Calculate the nth moment $E(X^n)$ for all n=1,... Express it in terms of the Gamma function.

I did try to integrate PDF to get distribution function, but I end up a constant $\frac{\alpha}{\beta}$, it must be wrong. Also, I have no idea about the second one. Can anyone please help me? Thanks a lot.

2. Originally Posted by tsang
Hi, I'm so struggling with this quetsion, can anyone please kindly help me? Thanks a lot.

Let $\alpha, \beta >0$ and
$f(x)=\left\{\begin{array}{cc}\frac{\alpha}{\beta}( \frac{x}{\alpha})^{\alpha -1} \exp(-(\frac{x}{\beta})^\alpha),&\mbox{ if } x>0\\0,&\mbox{otherwise}\end{array}\right$
and suppose the random variable X has density f.

(a)Determine the distribution function $F(x)$ of X
(b)Calculate the nth moment $E(X^n)$ for all n=1,... Express it in terms of the Gamma function.

I did try to integrate PDF to get distribution function, but I end up a constant $\frac{\alpha}{\beta}$, it must be wrong. Also, I have no idea about the second one. Can anyone please help me? Thanks a lot.
(a) Please show all your work so that it can be reviewed.

(b) Apply the definition and do the integration. Again, please show your work.

3. Looks like you need a beta where that alpha is in that ratio
Weibull distribution - Wikipedia, the free encyclopedia