Assume:

$ x | \theta \sim Poisson(\theta) $

1. What distribution would make a good (that is, derive the conjugate) prior? Show how you came to this conclusion, that is, do not just look it up.
2. Derive the posterior distribution and its parameters in this case.
3. Derive the marginal distribution of x.
4. Using a prior with shape=5 and inverse scale=5 and and data x=5. Compute the posterior. (These names are common names for the parameters to the distribution you should have found in number 1. Look up the parameter names on Wikipedia. But don't do it just to find out the distribution. It'll be obvious that you just looked it up anyway, if you don't have any explanation of why it's the correct prior.)

Change of Variables

One more Change of Variables