The number of storms in the upcoming rainy season is assumed to be Poisson distributed, but with a parameter $\Lambda$ that is also random and uniformly distributed on $(0,5)$. That is, $\Lambda \sim$ unif$(0,5)$ and given that $\Lambda = \lambda$, the conditional distribution of the number of storms $N$ is Poisson with mean $\lambda$: $N_{\mid\Lambda = \lambda} \sim$ Poisson ($\lambda$).
So far I've gotten (for the first parts of the question; not pertaining to the question below):
E$(N\mid \Lambda) = \lambda$
E$(N) = \frac{5}{2}$
Var$(N\mid\Lambda) = \lambda$
Var$(N) = \frac{55}{12}$
(It would be greatly appreciated if someone checked that)
Now what I need to answer is:
(i) Find the probability that zero storms occur this season (I think I need to integrate the pmf of Poisson distribution but I'm getting stuck)
(ii) Given that zero storms occur this season, what is the conditional distribution of $\Lambda$?
How do I go about answering these questions?
Best Answer
Since $N\mid\Lambda=\lambda \;\sim\;\mathcal{Pois}(\lambda)$ you know:
$$\mathsf P(N=0\mid \Lambda=\lambda) = \dfrac{\lambda^0 \mathsf e^{-\lambda}}{0!}$$
Since $\Lambda\sim\mathcal{U}(0;5)$ you also know $f_\Lambda(\lambda) = \frac 1 5 \mathbf 1_{\lambda\in(0;5)}$
And you should know how to find marginal distributions.
$$\mathsf P(N=0) =\int_0^5\mathsf P(N=0\mid\Lambda=\lambda) f_\Lambda(\lambda)\operatorname d \lambda$$
Put it together.
For (ii) use what you know about conditional probability (Bayes' Rule). The mix of probability densities and masses is not an issue for this question.
$$f_{\Lambda\mid N}(\lambda\mid n) \;=\; \frac{\mathsf P(N=n\mid\Lambda=\lambda)~f_{\Lambda}(\lambda)}{\mathsf P(N=n)}$$