So, there are 2 questions I have problems with answering and it would be great if anyone could point to where I go wrong...
1.Let $X_1, X_2, ..., X_n$ be i.i.d rvs with pdf $f(x|\theta) = (\theta+1)x^\theta, 0<\theta$.
Find the method of moments estimator of $\theta$
$$E(X) = \int_{0}^{1}x(\theta+1)x^\theta = (\theta + 1)\int_{0}^{1}x^{\theta+1}$$ $$E(X) = (\theta + 1)(\frac{x^{\theta+2}}{\theta+2}) \rvert^{1}_0 = \frac{\theta+1}{\theta+2}$$ $$\hat{\theta}_{MM} =...$$
2.Again, let $X_1, X_2,...,X_n$ be i.i.d rvs, with pdf $f(x|\theta)=3*\theta^{3}x^{-4}, \theta<x$
Find the mle of $\theta$
$$L(\theta|x_1, ..., x_n) = \prod_{i=1}^{n}3\theta^3x_i^{-4}$$ $$l(\theta) = \sum_{i=1}^{n}ln(3)+3ln(\theta)-4ln(x_i) = nln(3) + 3nln(\theta) -4\sum_{i=1}^{n}ln(x_i)$$ $$\frac{dl(\theta)}{d\theta} = \frac{3n}{\theta} = 0 $$ $$\hat{\theta}_{MLE} = ...$$
1) Just equate to $\bar{x}_n$ and put hats, i.e., $$ \bar{x}_n = \frac{\hat{\theta}_n + 1}{ \hat{\theta}_n + 2}, $$ then find $\hat{\theta}_n$, $$ \bar{x}_n\theta_n+\bar{x}_n2=\theta_n+1 \to \hat{\theta}_n=\frac{1-2\bar{x}_n}{-1+\bar{x}_n}. $$
2) Recall that $$ \hat{\theta}_{MLE} = \arg\max_{\theta \in \Theta} \mathcal{L} (\theta;x_1,...,x_n)\, . $$ Note that you should have the equality in $\ge$, otherwise you'll have to restate as the MLE as finding the supremum. Anyway, $$ \mathcal{L}{(\theta)} = 3 \prod \theta ^{x_i} \prod x_i^{-4}I\{x_i \ge\theta\}. $$ thus $x_{(1)} = \hat{\theta}_{MLE}$.