Question

3. You receive emails by a Poisson Arrival Process at a rate of 12 emails per hour.

(a) (6 points) Find the probability that you receive exactly 3 emails between 4:10 PM and 4:20 PM.

(b) (6 points) You start checking your email at 10:00 AM. What is the expected time of your first email?

(c) (9 points) Given that you receive exactly 10 emails between 4:00 PM and 5:00 PM, what is the (conditional) distribution of the number of emails you receive between 4:45 PM and 5:00 PM? For full credit, name the distribution and its parameters.

(d) (9 points) You read the emails you received between 10:00 AM and 11:00 AM and respond to them independently with probability 1/3.Let N be the number of emails you receive during that time window, and M be the number of emails you respond to. What is P(N= 0|M= 0)? (For full credit, your final answer should be in “closed form” and not include a summation.)

Answer 1

SOLUTION:

a> Let X be a random variable denoting the number of emails received between 4:10 pm and 4:20 pm.

X Poisson ( parameter = 12/60 = 0.2 )

Pr[X=3] = e-  x / x ! = e-0.2 0.23  / 3! = 0.0011 ( correct up to 4 decimal places )

The required probability is 0.0011

b> Let Y denotes the time to receive the first email.

Y Exponential ( with parameter = 0.2 )

Expected time for the first email = E[Y] = 1/0.2 = 5 minutes

c> Let N denotes the number of emails received between 4:00 pm and 5:00 pm and M denotes the number of emails received between 4:45 pm and 5:00 pm.

M|N=10 Binomial(n=10 , p = /n = 0.2/10 = 0.02 )

d> Response probability of an email : p = 1/3 = 0.3333

N| M   Binomial ( n= 0, p=0.3333)

Pr[ N=0 | M=0 ] = p ^0 (1-p) ^0

= 1