Question

Let X_{1}, X_{2}, . . . X_{n} be iid
exponential random variables with unknown mean β. Find the method
of moments estimator of β

Answer #1

We are given that X_{1}, X_{2}, . . .
X_{n} are i.i.d. exponential random variables with unknown
mean β. Thus, the population mean of X_{i}'s (i=1,2,...,n)
is given by:

Moreover, the sample mean is given by:

Now, under method of moments, we equate the sample moments to
the corresponding population moments to find the method of moments
estimator of unknown parameters (here the unknown parameter is β).
Since, we have to find the estimator of only one parameter, we
equate the sample mean to the population mean and we get:

Thus, the method of moments estimator of β is given by:

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