Recall from the slides that the Gamma distribution can be reparameterized using the two parameters a, the shape parameter, and \mu, the mean. The pdf looks like
\displaystyle \displaystyle f_{(a,\mu )}(y) \displaystyle = \displaystyle \frac{1}{\Gamma (a)}\left(\frac{a}{\mu }\right)^ a \, y^{a-1}\, e^{-\frac{ay}{\mu }}
Let {\boldsymbol \theta }=\begin{pmatrix} a\\ \mu \end{pmatrix}\, and rewrite this as the pdf of a 2-parameter exponential family. Enter {\boldsymbol \eta }({\boldsymbol \theta })\cdot \mathbf{T}(\mathbf{y}) below.
{\boldsymbol \eta }({\boldsymbol \theta })\cdot \mathbf{T}(\mathbf{y})=\quad
1 answer
{\boldsymbol \eta }({\boldsymbol \theta })\cdot \mathbf{T}(\mathbf{y})= \begin{pmatrix} \ln\left(\frac{a}{\mu}\right) \\ a-1 \end{pmatrix} \cdot \begin{pmatrix} y \\ 1 \end{pmatrix} = \ln\left(\frac{a}{\mu}\right) \cdot y + (a-1) \cdot 1 = y \ln\left(\frac{a}{\mu}\right) + a - 1
1 answer