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Let $X_1, \dots, X_n$ be i.i.d random variables with pdf: $f(x|\theta) = \theta x^2$, $0 < \theta \leq x < \infty$.

  1. Find the mle of $\theta$.

  2. Find the mle of $e^{\theta}$.

  3. Find the method of moments estimator of $\theta$.

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    Let me suggest (1.) to replace $\theta x^2$ by something which integrates to $1$, for example $\theta/x^2$, and (2.) to take @kahen's admonestation seriously.2012-09-30

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