Please help! I am very confused, and I want to make sure I have done these questions properly.

According to RIPTA, the 1 is supposed to arrive at the tunnel on Thayer 3 times between 1 pm and 2 pm on weekdays. Suppose you eat lunch at Starbucks every day and notice that, on average, the 1 really does come 3 times between 1 pm and 2 pm. However, due to weather and traffic conditions, the bus is rarely on time, and the actual number of times the bus comes is random. In this problem, we will show that the Poisson distribution is the natural probability distribution to describe how often the bus actually arrives. Let X be the random variable describing how many times the bus arrives at the stop between 1 pm and 2 pm on a specific day.

a) Let's assume that the bus is equally likely to arrive at any moment. What is the expected number of times the bus will arrive between 1 pm and 1:10 pm? How about 1 pm and 1:01 pm? Finally, what about 1 pm and 1:00:01 pm?

b) For each n = 1, ..., 3600, let X_n be the number of buses that arrive between n-1 and n seconds after 1 pm. Let's assume that we will not see two route 1 buses arriving within a second of one another so we can approximate each X_n with a Bernoulli random variable Ysubn with the same expectation. What is the parameter of each Ysubn?

c) Suppose the X_n's and Y_n's are independent. Given the assumptions above, how can we write X in terms of the X_n's? If Y = E3600 n=1 Y_n, then what is the distribution of Y? What are its parameters?

d) Find the expected value and variance of Y using your answer to part (c).

e) Use a calculator or R to compute the pig of Y up to 6 (that is, find P(Y = k) for k = 0, 1,..., 6). The Poisson distribution with mean has pmf P(x = k) = e^- (^k)/k!. Compare the pmfs of Y and the Poisson distribution with mean 3.

f) As we can see Y is approximately the Poisson distribution. If we replace seconds with milliseconds, the microseconds or even smaller intervals of time, then we get better and better approximations of the Poisson distribution. What assumptions have we made about how buses arrive in order to get this approximation?

g) If Z₁ and Z₂ are independent Poisson random variables with respective mean ₁ and ₂, then Z₁ + Z₂ is Poisson distributed with mean ₁ + ₂. Suppose the bus arrivals satisfy the assumptions you listed in part (f). In part (b), we assumed that no two route 1 buses would arrive in the same second to get Y as an approximation of X. However, if you take buses often, you may have seen two buses from the same route arriving together, so this is actually possible. What is the probability that two buses arrive in the same second sometime between 1 pm and 2pm?