Method of Moments (statistics) - Example

Example

Suppose X1, ..., Xn are independent identically distributed random variables with a gamma distribution with probability density function

for x > 0, and 0 for x < 0.

The first moment, i.e., the expected value, of a random variable with this probability distribution is

and the second moment, i.e., the expected value of its square, is

These are the "population moments".

The first and second "sample moments" m1 and m2 are respectively

and

Equating the population moments with the sample moments, we get

and

Solving these two equations for α and β, we get

\alpha={ m_{1}^2 \over m_{2} - m_{1}^2}\,\!

and

We then use these 2 quantities as estimates, based on the sample, of the two unobservable population parameters α and β.

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