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Let $Y_1, Y_2, . . . , Y_n$ be independent random variables, each with a beta distribution, with $α = β = 2$. Find a. the probability distribution function of $Y_(n)$ = max($Y_1, Y_2, . . . , Y_n$).

Could anyone help get me started on this?

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  • $\begingroup$ Hint: The event $\{\max_i Y_i \leq \alpha\}$ occurs if and only if $\cap_{i=1}^n \{Y_i \leq \alpha\]$ occurs. $\endgroup$ – Dilip Sarwate Feb 12 '14 at 5:55
  • $\begingroup$ Please use Y_{(n)}, not Y_(n). $\endgroup$ – Did Feb 12 '14 at 11:19
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Hint:

For i.i.d variables, $P(X_{(n)}<t)=(P(X<t))^{n}$.

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  • $\begingroup$ So, we can integrate f(y) for beta distribtion when $ α=β=2$ and raise it to the power of n? $\endgroup$ – afsdf dfsaf Feb 12 '14 at 6:05
  • $\begingroup$ Yes you can; but I should ask: What does this imply for the pdf of $X_{(n)}$? $\endgroup$ – Bombyx mori Feb 12 '14 at 6:12
  • $\begingroup$ we can find $f_u (t) = n[F_y(t)] ^{n-1} f_y(t) $ right? $\endgroup$ – afsdf dfsaf Feb 12 '14 at 6:15
  • $\begingroup$ Yes. Now you on the right track. $\endgroup$ – Bombyx mori Feb 12 '14 at 6:18
  • $\begingroup$ let $U = Y_{(n)}$ Is $F_u (t) = [3y^2 - 2y^3]^n$? $\endgroup$ – afsdf dfsaf Feb 12 '14 at 6:34

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