The number of emails a professor receives from students each hour on the day before an exam is a Poisson random variable. Recall that a Poisson random variable Y is the number of events that occur in some interval and that it has the following probability mass function: e-Pu p(y) = y = 1,2,3... y! Suppose that instead of being interested in the number of events (a discrete random variable), we are interested in the size of the interval between successive events (a continuous random variable). Let X denote the length of time from any starting point until an event occurs. 1. Assume that emails arrive at the rate of 1 = 0.1 emails per minute. What is the probability that the time between successive emails is at most 40 minutes? [Hint: If two events are successive then no other events occurred in the interval between them.] 2. Given a Poisson random variable Y with rate parameter 2 per unit interval, what is the probability that the interval between successive events, random variable X, is less than or equal to x? 3. What is the probability density function that corresponds to the cumulative distribution function you derived in question 2? The number of emails a professor receives from students each hour on the day before an exam is a Poisson random variable. Recall that a Poisson random variable Y is the number of events that occur in some interval and that it has the following probability mass function: e-Pu p(y) = y = 1,2,3... y! Suppose that instead of being interested in the number of events (a discrete random variable), we are interested in the size of the interval between successive events (a continuous random variable). Let X denote the length of time from any starting point until an event occurs. 1. Assume that emails arrive at the rate of 1 = 0.1 emails per minute. What is the probability that the time between successive emails is at most 40 minutes? [Hint: If two events are successive then no other events occurred in the interval between them.] 2. Given a Poisson random variable Y with rate parameter 2 per unit interval, what is the probability that the interval between successive events, random variable X, is less than or equal to x? 3. What is the probability density function that corresponds to the cumulative distribution function you derived in question 2