Typos per page with a weak Gamma prior

You model the number of typos per page in a manuscript as a Poisson process with unknown average rate $\lambda$ (typos per page). Your prior is $\text{Gamma}(\alpha=1, \beta=4)$, with shape $\alpha$ and rate $\beta$, a light belief centered at $0.25$ per page. A proofreader checks 10 pages and finds a total of 7 typos.

Using Bayesian updating, what is the posterior mean estimate of the typo rate $\lambda$?