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A marine biology research group is looking for a kind of precious aquatic plant in a particular sea area. Let X be the number of the plants per cubic kilometer in the sea area and assume X has the Poisson distribution Poisson(λ). Also, the parameter λ varies with the location and has a Gamma distribution with parameters α and β. (a) What is the expected number of plants per cubic kilometer? (b) What is the variance of the number of plants per cubic kilometer? (c) What is the probability they can find at least one plant per cubic kilometer?

For 1 c)

I got up to P(X>= 1) = 1- P(X=0) but am unsure of how to compute P(X=0)

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  • $\begingroup$ Not sure about the scope of the question but when $\lambda$ is constant, since $X \sim \mathcal{P}(\lambda)$, you have $$ \mathbb{P}[X=0] = e^{-\lambda} \frac{\lambda^0}{0!}=e^{-\lambda}. $$ $\endgroup$ – gt6989b Mar 27 '19 at 18:09
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I'll try to answer you question partially.

The idea behind the problem is the following. If you exactly know $\lambda$, you can say for sure that expected number of plants per cubic kilometer is $\lambda$.

However you don't exactly know $\lambda$, so the expected number of plants can be computed using the conditional probabilities. $E(X|\lambda = \hat{\lambda})= \hat{\lambda}$. $$E(X) = E(E(X|\lambda = \hat{\lambda}))$$ So you should take average over all possible averages (i.e. for all possible values of $\lambda$ counting(or averaging) of course with the pdf of $\lambda$).

So you will have $$E(X) = E(E(X|\lambda = \hat{\lambda}))=\int_{0}^{\infty}\hat{\lambda}f_{\lambda}(\hat{\lambda})d\hat{\lambda}=\int_{0}^{\infty}\hat{\lambda}\frac{\beta^{\alpha}}{\Gamma(\alpha)}\hat{\lambda}^{\alpha -1}e^{-\beta \hat{\lambda}}d\hat{\lambda}$$.

To our luck this is exactly the expectation of any gamma distribution. Which is $\frac{\alpha}{\beta}$.

For the variance and probability the idea in identical. (Try yourself using the comment above.).

However for more rigorous treatment of the problem look Poisson point processes, (This is another huge universe.) as your case is spatial.

Sorry if i didn't help.

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