481. The random variable $X$ (the water level in the river compared to the nominal level) is subject to a gamma distribution, the density of which is determined by the parameters $\alpha$ and $\beta$ $(\alpha>-1, \beta>0)$:

$$
f(x)=\frac{1}{\beta^{\alpha+1} \Gamma(\alpha+1)} x^{\alpha} \cdot \mathrm{e}^{-x / \beta} \quad(x \geqslant 0)
$$

Below is the distribution of the average water level based on data from $n=45$ floods (in the first row, the average water level $x_{i}$ (cm) is indicated; in the second row, the frequency $n_{i}$ - the number of floods with the average water level $x_{i}$ - is provided):

$\begin{array}{cccccccccc}x_{i} & 37.5 & 62.5 & 87.5 & 112.5 & 137.5 & 162.5 & 187.5 & 250 & 350 \\ n_{i} & 1 & 3 & 6 & 7 & 7 & 5 & 4 & 8 & 4\end{array}$

Find the point estimates of the unknown parameters $\alpha$ and $\beta$ of the considered gamma distribution using the method of moments.
Solution. We use the point estimates of the parameters of the gamma distribution (see problem 480):

$$
\alpha^{*}=\left(\bar{x}_{\mathrm{B}}\right)^{2} / D_{\mathrm{B}}-1, \quad \beta^{*}=D_{\mathrm{B}} / \bar{x}_{\mathrm{B}}
$$

From the given distribution, we can easily find the sample mean and sample variance: $\bar{x}_{\mathrm{B}}=166, D_{\mathrm{B}}=6782$.

Substituting these numbers into the formulas (*), we finally obtain the required point estimates of the unknown parameters of the considered gamma distribution: $\alpha^{*}=3.06, \beta^{*}=40.86$.