What is the CDF, pdf and expectation of $\theta^*$ of $\theta$?

Question

I have that $x_1, x_2,...,x_n$ are from a rv $X$ that has the density function $f_X(x)=\frac{2x}{\theta^2} \quad$ for $0 \le x \le \theta \quad$ and $f_X(x)=0 \quad$ otherwise. I have determined that $\theta^* = max(x_i)$. How Can I now calculate the CDF $F_{\theta^*}$, the pdf $f_{\theta^*}$ and the expectation $E[\theta^*]$ of the maximum likelihood estimator $\theta^*$?

Answer 1

Hints:

• If the $x_1,\ldots, x_n$ are independent, then $P(\theta^* \le t) = P(x_1 \le t, x_2 \le t, \ldots, x_n \le t) = \prod_{i=1}^n P(x_i \le t)$
• $f_{\theta^*}(t) = \frac{d}{dt} F_{\theta^*}(t)$
• Use the usual formula for computing expectation when you have the density function