If X∼Gamma(α,β)X \sim \text{Gamma}(\alpha, \beta)X∼Gamma(α,β) (with PDF βαΓ(α)xα−1e−βx\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}Γ(α)βαxα−1e−βx), what is E[1X]E[\frac{1}{X}]E[X1]?
βα−1\frac{\beta}{\alpha-1}α−1β
αβ\frac{\alpha}{\beta}βα
βα\frac{\beta}{\alpha}αβ
α−1β\frac{\alpha-1}{\beta}βα−1