# Method of Moment for Gamma Distribution

I know that the empirical $$r$$-th moment is defined as:

$$\hat E(X^r) = \frac{1}{n} \sum_{i=1}^n x_i^r$$

So for the first moment I did: $$E_{\lambda,\alpha}(X) = \hat E(X) = \bar X$$ $$\bar X = \frac{\alpha}{\lambda}$$ $${\alpha}=\lambda \bar X$$

For the second moment: \begin{align}E_{\alpha,\lambda}(X^2) & = \hat E[X^2] \\ & = \frac{(\alpha + 1) \alpha}{\lambda^2}\\ & = \frac{(\lambda \bar X + 1) \lambda \bar X }{\lambda^2}\\ & = \lambda \bar X^2 + \bar X = \lambda \frac{1}{n} \sum_{i=1}^n x_i^2 \end{align}

Know, this is where I´m stuck, I know for a fact, that the end equation must be, but I´m not sure how:

$$\lambda = \frac{\bar X}{ \frac{1}{n}\sum_{i=1}^n x_i^2 - \bar X^2}$$

• Please type your question as text, do not just post a photograph or screenshot (see here). When you retype the question, add the self-study tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. Apr 28, 2021 at 23:57
• @kjetilbhalvorsen How do I write math notation? And I don't get what else you want me to write, I just show you everything I did, the only thing they gave me was the equation that I had to get at the end Apr 29, 2021 at 0:09
• For writing math notation see math.meta.stackexchange.com/questions/5020/… Apr 29, 2021 at 0:10
• This is way wrong from the first line. E(X) is a parameter, and x-bar is a statistic. We use statistics to estimate parameters, but clearly understanding the distinction between statistics and parameters is fundamental to statistical thinking. Get that straight before trying to go any further. Apr 29, 2021 at 0:36
• The mean of the dats is not the same as the mean of the distribution. The latter is unvarying, but the mean of the data changes when you get a different set of data. Apr 29, 2021 at 2:30

In the sequence \begin{align}E_{\alpha,\lambda}(X^2) & = \frac{(\alpha + 1) \alpha}{\lambda^2}\\ & = \frac{(\lambda \bar X + 1) \lambda \bar X }{\lambda^2}\tag{1}\\ & = \lambda \bar X^2 + \bar X \tag{2}\\ & = \hat E[X^2] \\ & = \lambda \frac{1}{n} \sum_{i=1}^n x_i^2 \tag{3} \end{align} there is a mistake in (1), which makes (2) wrong. And there is an extra $$\lambda$$ in (3).