Poisson Thinning and Independence of Split Streams

Let $N \sim \text{Poisson}(\lambda)$. Each of the $N$ events is independently classified as type 1 with probability $p$ and type 2 with probability $1 - p$. Let $N_1$ and $N_2$ denote the counts of type 1 and type 2 events, respectively.

(a) Derive the marginal distribution of $N_1$.

(b) Derive the joint PMF $P(N_1 = j, N_2 = k)$ and show that $N_1$ and $N_2$ are independent.

(c) A website receives page views at rate $\lambda = 200$ per hour. Each visitor independently converts (makes a purchase) with probability $p = 0.03$. Find the probability of exactly 4 conversions in an hour, and the probability of at least 1 conversion given at most 210 total page views.