E(x)=∫(-∞，+∞)f(x)xdx=θ/( 1+θ)

x ' =σXi/n = E(x)=θ/( 1+θ)

θ=x'/( 1-x'), where σ xi/n

maximum likelihood estimate

f(xi.θ)=θ^n x 1^(θ- 1)x2^(θ- 1).xn^(θ- 1)

lnL(θ)= nlnθ+(θ- 1)ln(x 1x 2 . xn)

[lnL(θ)]'=n/θ+ln(x 1x2...xn)=0

θ=-n/ln(x 1x2.xn)

The maximum likelihood estimate is

θ=-n/ln(x 1x2.xn)