What distribution is result of division of two poisson distributed metrics?

Question

I've got two variables A and B with ≤ following a Poi(), resp. Poi() distribution.

Given values a1 and b1 we can figure out if these particular values are significantly decreased by calculating the probability of getting a1 and b2 or lower number:

where n=a and n=b, = and = b accordingly for each variable.

If the probability of getting a1(or b1) or lower value <5% threshold we can say that this particular value significantly decreased.

My question is how can we figure out the answer to the same question for variable C := A/B (so it is between 0 and 1). If we have got the value of c1 = a1/b2 then how do we know if c1 significantly decreased?

Answer 1

I suggest that, instead of $C:=A/B$, you consider random variable:

$$C:=B-A$$

In particular $C$ should be (or tested to be) Poisson-distributed with a parameter that you will call $\lambda c$. You should get $c:=b-a$ due to a version of the additive property of Poisson random variables. In this case, "significantly less" could be translated into "$C$ takes comparatively high values".

Remarks :

1) if sometimes $A > B$, you have no longer a Poisson distribution ; see in this case (Distribution of difference between independent Poisson random variables).

2) If you aren't convinced that it is better to study the difference, and keep trying to study the quotient $C:=A/B$, here is a reference that can be of interest.

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Edit : following our exchanges, if $C$ is defined by $C=B-A$, i.e. $B=A+C$, therefore $A/B=A/(A+C)=1/(1+C/A)$, I have done a simulation of Random Variable $R=A/B=1/(1+C/A)$ (see below)

Taking $A$ and $C$ both Poisson with resp. parameters $\lambda=20$ and $\lambda=25$, we obtain a rather gaussian distribution. Here is the Matlab program that has generated this histogram:

function ratpoi;
for q=1:100000
   A=poi(20);C=poi(25);
   T(q)=1/(1+C/A); 
end;
hist(T,30);

function P=poi(L); % gen. Poisson Rand. Var. with par. L
EL=exp(-L);k=0;p=1;
while p>EL
    p=p*rand;k=k+1;
end
P=k-1;