2. (0.75pt) Consider a Poisson process whose average rate is \lambda arrivals per minute. If there is only one arrival in the first minute, find the distribution of the time of that arrival.
("Find the distribution" means "find the name of the type of random variable and its parameters if it is a famous random variable, and find its pdf if not.")
. Answer: Uniform(0, 1), which is the same thing as Beta(1, 1). In particular, all "\lambda"s should cancel out.
Hints: I thought of two possible methods to solve this question.
Method 1: This problem asks for a distribution of one RV, given that another RV equals a number. What are those two random variables? Are they discrete? Continuous? One of each? Bayes's theorem is a common way to find find that kind of distribution, but make sure you use the right version.
Method 2: Let Y be the time of the first arrival, and X be the time between the first and second arrival. Then having one arrival in the first minute is the same as Y < 1 < X + Y (why?). That means this question is asking for the distribution of Y given that Y < 1 < X + Y. Abbreviate "Y given Y < 1 < X + Y" as Z. Then Z is a continuous RV, so one way you can find its distribution is to compute its cdf $F_Z(z)$.
