Question

Suppose X is a positive random variable with density $f_{x}(x)$ for x > 0 and with moment generating function $M_{x}(t).$

(a) Use the fact that

$x^{-1} = \int_{-\infty }^{0}e^{ux}du$

for x > 0 to prove that

$Expected Value(x^{-1}) = \int_{0 }^{\infty}M_{x}(-t)dt$

(b) Use the result in (a) to find ExpectedValue(x^-1) if X ~ Gamma(alpha, beta) where the ExpectedValue(X) = alpha(beta)

Answer 1