Consider a gamma distribution $\Gamma (\alpha,\beta)$ and $X = \{1.49,6.3,6.41,1.07,1.0,4.4,3.05,2.23,1.65,5.75\}$. Assume $\alpha = 3.5$, what's the MLE of $\beta$?
I started to calculate $$L(\beta | X)= \prod_{i=1}^{n}\frac{\beta^{\alpha}}{\Gamma(\alpha)}X_i^{\alpha-1}e^{-\beta X_i}$$
and by working with the logarithm I confirmed the maximum. $g_{\beta} = \log(L(\beta|X))$
\begin{align*} &\frac{\partial g_{\beta}}{\partial \beta}=n\frac{\alpha}{\beta}-\sum_{i=1}^n X_i \Rightarrow \beta = \frac{\alpha}{\bar{X}}\\ &\frac{\partial^2 g_{\beta}}{\partial \beta^2} = -n\frac{\alpha}{\beta^2} <0 \end{align*}
So my solution is $\beta = \frac{35}{34.25}$ but the solution from the exercise is $\beta = \frac{34.25}{35}$. Did I forget something or is the example solution wrong?