Gamma density ($f(y)={\theta }^k{y}^{k-1}{e}^{-\theta y}/\Gamma (k),y>0$)

The probability density function of a Gamma distribution with shape $k>0$ and rate $\theta >0$. Integration against $y^{-j}$ gives $\mathbb{E}[Y^{-j}]={\theta }^j\Gamma (k-j)/\Gamma (k)$ when $j<k$; the integral diverges for $j\ge k$. Rate and scale are reciprocal conventions and must be distinguished.