Let $N$ be the number of requests to a web server per day and let $N \sim \mathrm{Poisson}(λ)$. Each request $5$. Let each request come from a human with probability $p$ or from a spam bot with probability $1 − p$. Assume that the requests are independent of each other. Let $X$ be the number of requests from humans per day and $Y$ be the number of requests from spam bots per day
(a) State the conditional distribution of $X$ given $N = n$, and state the conditional distribution of $Y$ given $N = n$.
(b) Calculate the probability of getting exactly $x$ human requests and $y$ spam requests.
For a) I got $$P(x\mid N=n) = {n \choose x} p^x (1-p)^{n-x}$$
Am I on the right path?
c) Find the joint probability mass function of X and Y , and show that X and Y are independent of each other.
im not sure how to get the joint pmf
Your answer for Part A looks fine to me.
Hint For Part B
You are being asked to find $\mathbb{P}(X=x \land Y=y)$
Something else to point out is that $N = X+Y$
The probability can then be found as follows:
$$\mathbb{P}(N=x+y)*\mathbb{P}(X=x \mid N=x+y)$$
Can you take it from here? (The $Y$'s don't need to be solved as they will have been determined already once you have solved it for $x$)