1. ## Order statistics help

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

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

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

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

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

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

Thus, $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
.

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

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

Thanks

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

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

$F_{Y_{(n)}}(y)=P(Y_{(n)}\theta \end{cases}$

CB

3. Just because you can't recognize this, doesn't mean it's wrong.
AND there is only one y, whether you call it $y_{k}$ or $y_{n}$.....

$\frac{n}{\theta}[\frac{y}{\theta}]^{n-1}, 0

is the derivative of the CDF that CB derived.

If you are curious....
If we let $X=Y/\theta$ then X is a Beta and you can obtain the mean and variance that way if you wish.

4. Originally Posted by matheagle
If you are curious....
If we let $X=Y/\theta$ then X is a Beta and you can obtain the mean and variance that way if you wish.
I am not seeing how X is a beta random variable. What would my alpha and beta be?

5. Originally Posted by Danneedshelp
I am not seeing how X is a beta random variable. What would my alpha and beta be?
$\beta=1$ and $\alpha=N$

CB