10-58. Method of Moments Technique for Finding Point Estimators:

Let {X1,X2, . . . ,Xn} be a set of samplerandomvariables taken from a distribution with a probability density function f (x; θ1, . . . , θk). Let μ



r

= E(Xr)

denote the rth moment of the distribution withM

r

= ni

=1 Xr i

n representing the rth moment of the sample, r = 1, 2, 3, . . . . Then the method of moments has us setE(Xr) toM

r for as many values of r = 1, 2, . . ., as needed in order to obtain enough equations to uniquely solve for θ1, . . . , θk. For X1, . . . ,Xn taken from f (x; θ) = θxθ−1, 0 < x < 1; 0 < θ < +∞, obtain an estimate of

θ by the method of moments. (Here we need only r = 1.)