I think I've solved part a, but would like confirmation, not 100% sure.
For part b, I'm pretty lost. I think it's a poisson distribution because it's modeling wait time? But at the same time, it has frames between arrivals, so I'm not sure.
Let $X_t$ be the number of arrivals after minutes in a process with arrival rate = $4.2~ \text{min}^{-1}$ . Use a Binomial process with $2$-second frames to model $X_t$ .
(a) Identify the distribution of $X_t$ and compute $(X_t \leq 50)$ using the Central Limit Theorem.
$\delta = 2/60$
$n = t/\delta = 15/.03333 = 30$
$p = \delta\times\lambda = 4.2 \times.0333$
$X\sim \text{Binomial}(n=30, .14)$
$P(X < 50) = \text{ Binomial CDF } (30, .14, 50)$
(b) Identify the distribution of the inter-arrival time (by relating it to frames), then compute the probability that 30 seconds pass before the next arrival.
No, it is not Poisson.
Hint: You are modelling arrival in a 2second frame as Bernoulli trials, so the inter arrival time is count of trials until the next success (multiplied by frame length).
What distribution is had by the count of trials until the next success in a sequence of Bernoulli trials?