If X∼Gamma(α,β)X \sim \text{Gamma}(\alpha, \beta)X∼Gamma(α,β) with PDF f(x)=βαΓ(α)xα−1e−βxf(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}f(x)=Γ(α)βαxα−1e−βx, what is the value of E[ln(X)]E[\ln(X)]E[ln(X)]?
ψ(α)−ln(β)\psi(\alpha) - \ln(\beta)ψ(α)−ln(β)
ψ(α)+ln(β)\psi(\alpha) + \ln(\beta)ψ(α)+ln(β)
ln(α)−ψ(β)\ln(\alpha) - \psi(\beta)ln(α)−ψ(β)
1α−β\frac{1}{\alpha} - \betaα1−β