480. Find the method of moments estimates for the unknown parameters $\alpha$ and $\beta$ of the gamma distribution from the sample $x_{1}, x_{2}, \ldots$, $x_{n}$, where the density is given by

$$
f(x)=\frac{1}{\beta^{\alpha+1 \Gamma}(\alpha+1)} x^{\alpha} e^{-x / \beta} \quad(\alpha>-1, \beta>0, x \geqslant 0)
$$
Solution. To find two unknown parameters, it is necessary to have two equations; equate the initial theoretical moment of the first order $v_{1}$ to the initial empirical moment of the first order $M_{1}$ and the central theoretical moment of the second order $\mu_{3}$ to the central empirical moment of the second order $\boldsymbol{m}_{2}$;

$$
\boldsymbol{v}_{\mathbf{1}}=\boldsymbol{M}_{1}, \boldsymbol{\mu}_{2}=\boldsymbol{m}_{\mathbf{2}}
$$

Considering that $v_{1}=M(X), M_{1}=\overline{x_{B}}, \mu_{2}=D(X), m_{2}=D_{B}$, we have

$$
\left\{\begin{array}{l}
M(X)=\bar{x}_{\mathrm{B}}, \\
D(X)=D_{\mathrm{B}}
\end{array}\right.
$$

The mathematical expectation and variance of the gamma distribution are respectively equal to $M(X)=(\alpha+1) \beta, D(X)=(\alpha+1) \beta^{2}$ (see problem 302), therefore (") can be written as

$$
\left\{\begin{array}{l}
(\alpha+1) \beta=\bar{x}_{\mathrm{B}} \\
(\alpha+1) \beta^{2}=D_{\mathrm{B}}
\end{array}\right.
$$

Solving this system, we finally obtain the desired point estimates of the unknown parameters: $\alpha^{*}=\left(\overline{x_{\mathrm{B}}}\right)^{2} / D_{\mathrm{B}}-1, \beta^{*}=D_{\mathrm{B}} / \overline{x_{\mathrm{B}}}$.