The number of emails per day in Matt's inbox can be modeled by a Poisson random variable with mean=30.

(a)Suppose the random variableXequals the number of emails in 1 hour. Determine the distribution ofXand findE(X)andSD(X).

(b)What is the probability that Matt receives 3 emails in one hour?

(c)What is the probability that Matt receives 1 or more emails in one hour?

(e)Suppose Matt's inbox is watched over a 2 day period. Find the probability that 20 emails arrive in the first day and 30 emails arrive in the second day.Hint: Remember that the number of occurrences in non-overlapping intervals in a Poisson process are independent. If we letX1andX2equal the number of emails in the first and second day, respectively, then we can findP(X1=20 andX2=30).

(d)Suppose Matt's inbox is watched over a 3 day period. Find the probability that 20 emails arrive in the first day and 50 emails arrive in the last 2 days.Hint: If we letX1equal the number of emails in the first day, and letX2equal the number of emails in the last 2 days, then we can findP(X1=20 andX2=50).

(e)Suppose we observe 24 one-hour time intervals. Determine the probability that Matt receives exactly 1 email in 3 of the intervals.Hint: First determine the probability of 1 email in 1 hour, then make use of the binomial distribution.