This is the problem. Two roads A and B lead to a motel. The number of clients who come to the motel per day on each road are independent Poisson variables X and Y with parameters r and s respectively. It is known that the total number of clients who attended one day it was n.
a) Find the probability that the number of customers who arrived by road A were k with 0 ≤ k ≤ n
My best approach has been thinking in its their joint probability, and since they are independent it would only be the multiplication of the probabilities.
Then my answer would be $P(X<=k)$ multiplied by $P(Y<=n-k)$ because if $X<=k$ then $Y$ can't be greater than $n-k$ since the total of arrives was $n$, therefore final answer would be:
$$P(X <= k, Y<=n-k)$$
But I fell that is wrong. And my other approach is the sum of $X$ and $Y$, but I don't see much sense doing that.
Someone have any idea?
Sorry if any idea is not understood.
Thanks.