Expected value of the Max of IID random variables that follow the Discrete Uniform Distribution

Question

I'm trying to find the expected value of $X_{n}$, where $X_{n}$ is the MAX of {$X_{i}$, ..., $X_{n}$} and X ~ U(0, 2Θ), with Θ > 0. I don't know if what i'm doing is right, but so far what I got is; $$P(X_{n}\le x)=P(X_i \le x ,i=1,2,...,n)$$ $$CDF = P(X_{n}\le x)=\prod_{i=1}^{n} P(X_i\le x)=x^{n}$$ $$PDF = \frac{d}{dx}(x^{n}) = nx^{n-1}$$ $$E[X]=\int _0^{2Θ} x (nx^{n-1})dx=\int _0^{2Θ} nx^{n}dx=\frac {n2^{n+1}Θ^{n+1}}{n+1}$$

Answer 1

In general your way of solving the problem is correct, but you have some problems:

1. You use "$X_n$" for both the last of the uniform variables and also for the max of them all. Please change your notation for the latter.
2. The probability of the individual uniform variables has the wrong normalization. The probability of $X_i\le x$ is not $x$, but $x/2\theta$