# Find the probability distribution function of $Y_(n)$ = max($Y_1, Y_2, . . . , Y_n$).

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?

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

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