problem 14

Let $F$ be the CDF and $f$ the PDF of $X$ .

The maximum is less than or equal to some $x$ if and only if all $U_{i}$ are less than or equal to $x$ .

P (X \leq x) = P (U_{1} \leq x, U_{2} \leq x, \dots, U_{n} \leq x), for 0 < x < 1

Since the r.v.s $U_{i}$ are independent, the events ${U_{i} \leq x}$ are independent as well.

P (X \leq x) = P (U_{1} \leq x) P (U_{2} \leq x) \dotsP (U_{n} \leq x)

The CDF of the standard uniform is equal to $x$ for $0 < x < 1$ . Thus:

F (x) = P (X \leq x) = {\begin{matrix} 0 & , for x \leq 0 x^{n} & , for 0 < x < 1 \end{matrix}

The PDF is equal to the derivative of the CDF

f (x) = F^{'} (x) = n x^{n - 1}, for 0 < x < 1

By the definition of expectation for continuous r.v.s

E (X) = \int_{0}^{1} xf (x) dx = \int_{0}^{1} n x^{n} dx = \frac{n}{n + 1}