Q: Let $\displaystyle Y_{1},...,Y_{n}$ by independent, uniformly distributed random variables on the interval $\displaystyle [0,\theta]$. Find the

a) Probability distribution function of $\displaystyle Y_{(n)}=max(Y_{1},Y_{2},...,Y_{n})$.

b) density function of $\displaystyle Y_{(n)}$.

c) mean and variance of $\displaystyle Y_{(n)}$.

A: If I can figure out a) and identify the pdf, the cdf (part b) and mean / variance (part c) should follow. So, here is my attempt

Let $\displaystyle Y_{(n)}=max(Y_{1},Y_{2},...,Y_{n})$. Then, by Thereom blah, we have

$\displaystyle f_{Y_{(n)}}=g_{(n)}(y_{n})$$\displaystyle =

\frac{n!}{(n-1)!(n-k)!}[F(y_{k})]^{k-1}[1-F(y_{k})]^{n-k}f(y_{k})$,$\displaystyle -\infty<y_{k}<\infty$ with $\displaystyle k=n$.

Thus, $\displaystyle f_{Y_{(n)}}=g_{(n)}(y_{n})=

\frac{n!}{(n-1)!(n-n)!}[\frac{y_{n}}{\theta}]^{n-1}[1-\frac{y_{n}}{\theta}]^{n-n}

\frac{1}{\theta}=

\frac{n}{\theta}[\frac{y_{n}}{\theta}]^{n-1}, 0<y_{k}<\theta$.

I do not recognize the density function $\displaystyle \frac{n}{\theta}[\frac{y_{n}}{\theta}]^{n-1}, 0<y_{k}<\theta$ and its support.

Any help would be great. Not sure where I went wrong.

Thanks