Stats posterior probability gamma conjugate family

[binbagsss]

Question

Find the posterior probability that the next two observations y4 and y5 will both be zero? Where the prior distribution is a gamma with parameters (a,b) and the sample is of size of 3 taking from a poisson disribution with parameter V.

So far I have shown that the posterior distribution is also gamma with parameters a*=a+t, where t=y1+y2+...yn, and b*=b+n, n the sample size.

Attempt at the solution

In my notes, a different question but all i have on my notes on this- is that the probability of exactly the next observation being 1 is given as:
$P(y=1 | D)= \int^{\infty}_{0} P(y=1|V) P(V|D) dV,$ So wheree P(V|D) is the updated posterior gamma distribution,

where D is the data, the observations y1+...+yn.

This is for one sample and I'm unsure how to approach for 2 samples. The ideas I have are:
1) Compute $P(y=0 | D)= \int^{\infty}_{0} P(y=1|V) P(V|D) dV$ , and then since we want consecutvie 0 to just square this.
2) To compute $P(y4=0 an y5=0)=\int^{\infty}_{0} P(y4=1 and y5=1|V) P(V|D) dV$ , so i.e. is proportional to $e^{-2V}$ ,

Are any of these methods correct? Could someone please explain which is right or wrong, or if both or wrong and what I should be doing,

Thanks for any help, really appreciated.

 

[binbagsss]

bump.