Find the method of moments estimator

Let $X_1, X_2, ..., X_n$ be a random sample from a distribution who's PDF is given by $f(x; \theta)=(\theta+1)x^{\theta}$ for $0 \leq x \leq 1$ or $0$ otherwise.

Find the method-of-moments estimator for $\theta$.

So I have done the following:

$\mathbb{E}(X)=\int_{-\infty}^{\infty} xf(x) dx = (\theta +1)\int_{0}^{1}x^{\theta+1}dx=\frac{\theta +1}{\theta+2}$

I am unsure now how to equate this to $\bar{X}$

• Seems like all you need to do is just to equate it to the sample mean $\frac{\theta + 1}{\theta +2} = \frac{1}{n}\sum_{n=1}^n X_i$ and solve the equation for $\theta$ (the obtained value will be the estimator $\hat{\theta}$) – applyb Aug 25 '16 at 19:17
• You simply equate corresponding population and sample moments to obtain the estimator for $\theta$. – Glen_b Aug 31 '16 at 3:01

It's like Beck says. $\theta +1=\bar X (\theta+2) \Leftrightarrow \theta=\frac{2\bar X -1}{1-\bar X}$