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- Jan 26, 2012

- 995

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**Problem**: Let $X_1,X_2,\ldots,X_n$ be independent random variables, each having a uniform distribution over $(0,1)$. Let $M=\max(X_1,X_2,\ldots,X_n)$. Show that the distribution function of $M$, $F_M(\cdot)$, is given by \[F_M(x)=x^n,\qquad 0\leq x\leq 1.\]

What is the probability density function of $M$?

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