Show that expression is Gamma distribution with given parameters
Solution 1:
We have
$$f(\lambda) = \frac{n^k}{p(y)}\lambda^ke^{-n\lambda}$$
is a density function where $\frac{n^k}{p(y)}$ is independent of $\lambda$.
Hence, by comparing with the pdf of gamma distribution, we conclude that it is a gamma distribution with the shape parameter being $k+1$ and the corresponding rate is $n$.
Note that the shape is equal to
$$k+1 = \sum_{i=1}^n y_i + 1 = n\bar{y} + 1.$$