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Math Help - Order statistics help

  1. #1
    Senior Member Danneedshelp's Avatar
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    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}) =<br />
\frac{n!}{(n-1)!(n-k)!}[F(y_{k})]^{k-1}[1-F(y_{k})]^{n-k}f(y_{k}),  -\infty<y_{k}<\infty with k=n.

    Thus, f_{Y_{(n)}}=g_{(n)}(y_{n})=<br />
\frac{n!}{(n-1)!(n-n)!}[\frac{y_{n}}{\theta}]^{n-1}[1-\frac{y_{n}}{\theta}]^{n-n}<br />
\frac{1}{\theta}=<br />
\frac{n}{\theta}[\frac{y_{n}}{\theta}]^{n-1}, 0<y_{k}<\theta.

    I do not recognize the density function \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
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Danneedshelp View Post
    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)}<y)=\prod_{i=1}^n P(Y_i<y)=\begin{cases}0&,\ y<0 \\ \left(\frac{y}{\theta}\right)^n & ,\  0 \le y \le \theta \\1 & , \  y>\theta \end{cases}


    CB
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  3. #3
    MHF Contributor matheagle's Avatar
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    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<y<\theta

    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.
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  4. #4
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by matheagle View Post
    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?
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by Danneedshelp View Post
    I am not seeing how X is a beta random variable. What would my alpha and beta be?
    \beta=1 and \alpha=N

    CB
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