Question

8. Let (x1, x2, ..., xn) be independent measurements of a random variable X with density function

f(x) = e−(x−α), x > α. Find an estimator, ˆ

α, of α by method of moments.

Answer 1

$EX_1=\int_a^{+\infty}{xf\left( x \right) dx}=\int_a^{+\infty}{xe^{-x+a}dx}=e^a\left( -xe^{-x}|_{a}^{+\infty}+\int_a^{+\infty}{e^{-x}dx} \right) =\\=e^a\left( ae^{-a}+e^{-a} \right) =a+1\\\bar{x}=\hat{a}+1\Rightarrow \hat{a}=\bar{x}-1$