Probability – Expected Value of Max of Uniform IID Variables

Question

What is the expected value of the maximum of 500 IID random variables
with uniform distribution between 0 and 1?

I'm not quite sure of the technique to go about solving something like this. Could anyone point me the right direction?

Thanks

Answer 1

Suppose the maximum is $X_{500}$, then $$P(X_{500}\le x)=P(X_i \le x ,i=1,2,...,500)$$ Note that this is so because if the maximum is less than $x$ , then every other order statistic is less than $x$. Now since the $X_i's$ are IID, it follows that; $$P(X_{500}\le x)=\prod_{i=1}^{500} P(X_i\le x)=x^{500}$$ which is the CDF and so the PDF is $500x^{499}$ (which is obtained by differentiation). Now the expected value of the maximum is found as follows; $$E[X]=\int _0^1 x (500x^{499})dx=\int _0^1 500x^{500}dx=\frac {500}{501}$$