Question: Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with the following

p.d.f.

\(f(x)= \begin{cases}\frac{3}{\theta^{3}} x^{2}, & \text { if } 0 \leqslant x \leqslant \theta ; \\ 0, & \text { otherwise. } \end{cases}\)

1. Find the method of moments estimator \(\tilde{\theta}\) of \(\theta\).
2. Find \(E(\tilde{\theta})\). Is \(\tilde{\theta}\) an unbiased estimator of \(\theta ?\) Explain why.
3. Find \(\operatorname{Var}(\tilde{\theta})\) and \(\lim _{n \rightarrow \infty} \operatorname{Var}(\tilde{\theta})\).
4. Find the maximum likelihood estimator $\hat{\theta}$ of $\theta$.

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