Hi. I normally can solve poisson distribution questions with ease. But this one question had me thinking for hours on end with no solution. It would be great if someone can help me.
QN:
The number of incoming calls per minute, X, to a telephone exchange has a Poisson distribution with mean 2. The number of outgoing calls per minute, Y, has an independent Poisson Distribution with mean 1.
In a randomly chosen 1-minute period, find the probability that there are more incoming calls than outgoing calls, given that there is a total of AT LEAST 3 calls.
My Working:
Let incoming calls > outgoing calls be A
Let a total of at least 3 calls be B
P(A|B) = P(A∩B)/P(B)
The part where i am stuck is finding P(A∩B), i.e, Probability of more incoming calls than outgoing calls AND a total of at least 3 calls. There are so many cases. Could be 3 incoming calls, 0 outgoing calls. Could be 4 incoming calls 1 outgoing calls. E.t.c. Then you take the sum of all these cases and it would be = P(A∩B).
But the number of cases are endless. So, just how am i supposed to calculate P(A∩B)? Im at my wits ends!
The answer is 0.705 if you manage to solve it.
QN:
The number of incoming calls per minute, X, to a telephone exchange has a Poisson distribution with mean 2. The number of outgoing calls per minute, Y, has an independent Poisson Distribution with mean 1.
In a randomly chosen 1-minute period, find the probability that there are more incoming calls than outgoing calls, given that there is a total of AT LEAST 3 calls.
My Working:
Let incoming calls > outgoing calls be A
Let a total of at least 3 calls be B
P(A|B) = P(A∩B)/P(B)
The part where i am stuck is finding P(A∩B), i.e, Probability of more incoming calls than outgoing calls AND a total of at least 3 calls. There are so many cases. Could be 3 incoming calls, 0 outgoing calls. Could be 4 incoming calls 1 outgoing calls. E.t.c. Then you take the sum of all these cases and it would be = P(A∩B).
But the number of cases are endless. So, just how am i supposed to calculate P(A∩B)? Im at my wits ends!
The answer is 0.705 if you manage to solve it.