Part 1: Distribution of and Conditional Probability
Given that and , and they are independent:
Distribution of :
The sum of two independent Poisson random variables is also Poisson-distributed, with the rate parameter being the sum of their rate parameters:
Conditional Probability :
Given , the conditional distribution of is Binomial with parameters and . That is:
for .
Part 2: Distribution of
, :
The pmf of a geometric random variable is:
Distribution of :
For , the event occurs if both and . Since and are independent:
For geometric random variables:
Thus:
The pmf of is:
Hence, .
Part 3: Probability
The event occurs with probability:
since and are independent:
Thus:
This is a geometric series with ratio , so:
Part 4: Joint pmf of
Let . Then , and represents the smaller value of and .
Joint pmf :
For and , we have:
- , , or
- , .
The joint pmf is:
Using the pmf of geometric distributions:
Similarly, for the second term:
Thus:
Factoring terms: