I am stuck with some parts of a problem in my textbook, and the solutions in my textbook do not seem to help me. The problem goes:
Two independent observations $X_1$ and $X_2$ are made of continuous random variables with pdf: $$f(x)=\frac{1}{k}; 0\le x \le k$$
i) How do I find a probability distribution of M which stands for the larger of $X_1$ and $X_2$?
ii) how do I show that M is an unbiased estimator of $k?$
iii) how does $X_1 + X_2$ for an unbiased estimator of $k?$
My thoughts:
I think I know how to do the unbiased estimator problems as you just have to put in the estimator in to $E(X)$ and show it equals to whatever the parameter may be. However I am confused to what the parameter here is to begin with, are they assuming $k $is the parameter? So for the problem iii), do I go $E(X_1+X_2)=E(X_1)+E(X_2)$ and use the mean results from continuous uniform distributions?
The question i) and ii), I am completely stuck, I do not know where to go with this problem.
I would appreciate the help offered.
i) Your variables are uniform on $[0,k]$. You can see this from the density you wrote.
ii) $M = max\{X_1, X_2\}$ is not an unbiased estimator for $k$. To see this you need to obtain $M$'s density and from there calculate E(M). You can derive the density from the fact that $F_M(x) = P(X_1 \leq x, X_2 \leq x) = \frac{x}{k}\frac{x}{k}$. $F_M(x) = \frac{x^2}{k^2} \Rightarrow f_M(x) = 2\times \frac{x}{k^2} \Rightarrow E(M) = \int_0^k2\times \frac{x^2}{k^2}dx= \frac{2}{3}k.$
iii) $E(X_1 + X_2) = E(X_1) + E(X_2) = \frac{k}{2} + \frac{k}{2} = k$ I.e. $X_1 + X_2$ is an unbiased estimate for $k$.