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. Ihave to determine the MLE of $\theta^*$ of $\theta$
Here is how I have done it:
$L(\theta)= \frac{2}{\theta^{2n}}\prod_{i=1}^nx_i$
$\frac{\partial L(\theta)}{\partial \theta} =...=\frac{-4n}{\theta^{2n+1}}\prod_{i=1}^nx_i + \frac{2}{\theta^{2n}}\frac{\partial(\prod_{i=1}^nx_i)}{\partial \theta}$
Is this correct? and also how do I calculate the CDF $F_{\theta^*}$, the pdf $f_{\theta^*}$ and the expectation $E[\theta^*]$ of the maximum likelihood estimator $\theta^*$?
What is the MLE $\theta^*$ of $\theta$?
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statistics
statistical-inference
maximum-likelihood
cumulative-distribution-functions
robust-statistics