Question

Let X1,x2, x3...xn, be independent random variables each having uniform distribution ocwe (0,1). Let M=max(x1...xn). show that the distribution of M, Fm(x) is given by x^n

Answer 1

\[F_M(x)=P(M\leq x)=P(\max(x_1,x_2,\ldots,x_n)\le x)\]

\[=P(x_1\le x,x_2\leq x,\ldots,x_n\le x)=\prod_{i=1}^nP(x_i\le x)\]

\[=\prod_{i=1}^nx=x^n\]

Answer 2

thanks for your answer which looks right. but could you please dumb it down a bit for me. how do you get from the last but one step to the final result. i missed that.

Sorry to have you oversimplify. Thanks

Answer 3

\[\text{all the } x_i \text{ are uniform (0,1) so for any number }a \text{ with }0\le a\le 1\text{ we have}\]

\[P(x_i<a)=\int\limits_{0}^{a}1dx=a\]

so \[P(x_i<x)=x\text{, provided that } 0\le x\le 1\]