Hi

I'm trying to work out the expected value of the continuous random variable Y:

$\displaystyle E[Y]=\int_{0}^{\infty}\frac{n{}^{n}\lambda{}^{n}}{\Gam ma(n)}y^{n-1}e^{-ny\lambda}\textrm{d}y=\frac{n{}^{n}\lambda{}^{n}}{ \Gamma(n)}\int_{0}^{\infty}y^{n-1}e^{-ny\lambda}$

note: in the denominator right after the first integral sign, it should read Gamma(n) not ma(n). not sure what happened to me TeX code.

where n is a positive integer constant, and lambda is a constant

so far i've tried integration by parts, but i got a recursion

Integration by parts

let $\displaystyle u=y^{n-1}$ and $\displaystyle v'=e^{-ny\lambda}$

$\displaystyle u'=(n-1)y^{n-2}$$\displaystyle and v=\frac{-1}{n\lambda}e^{-ny\lambda}$

$\displaystyle I=uv-\int vu'\textrm{d}y$

$\displaystyle =\left[y^{n-1}\frac{-1}{n\lambda}e^{-ny\lambda}\right]_{0}^{\infty}-\int_{0}^{\infty}(n-1)$