Method of moments for a beta fill-rate model

Question

The fraction of an order that gets filled each day is a number between $0$ and $1$ , which you model with a Beta $(\alpha, \beta)$ distribution. From your data the sample mean is $0.4$ and the sample variance is $0.04$ .

Use the method of moments to estimate $\alpha$ and $\beta$ .

Accepted Answer

Two parameters, match mean and variance. For a Beta(\alpha, \beta) distribution the mean and variance are \mathbb{E}[X] = \frac{\alpha}{\alpha + \beta}, \qquad \operatorname{Var}(X) = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}. Call the sample mean m and the sample variance v. Rather than wrestle with those formulas directly, use the standard rearranged result \alpha + \beta = \frac{m(1 - m)}{v} - 1. This "total concentration" tells you how tightly the distribution clusters: smaller variance means a larger \alpha + \beta. Plug in the numbers. With m = 0.4 and v = 0.04, first compute…

Method of moments for a beta fill-rate model

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