The number of emails that I get in a week day is (Monday - Friday) can be modele

Question

The number of emails that I get in a week day is (Monday - Friday) can be modeled by Poisson distribution with average of 1/6 emails per minute. The number of emails I recieve on weekends (Sat, Sun) can be modeled by a Poisson distribution with an average of 1/30 emails per minute.

(A) What is the probability that I will get no emails in an interval length 4 hours on Sunday?

(B) What is the distribution of the number of emails I get over one entire week. Beginning Monday to end on Friday?

(C) Find the distribution of the number of emails I will get over one entire week compute the average of emails I get over a week.

(D) A Random day is choosen (all days of the week are equally likely to be selected) and a random interval of length one hour is selected on the chosen day. It is observed that I recieced 5 emails in that interval. What is the probability that the chosen day is a weekday?

Explanation / Answer

Expected mails on a weekend day =(1/30)*60*24 =48

a)Expected mails in 4-hours on weekend day = (1/30)*60*4 =8

P(n=0) = e-8 80 /0! = 3.35 * 10-4

c)Average no of emails recieved in week =7*(1/6)*24*60 =1680

d)expected emails in one hour on week day=(1/6)*60 =10

expected emails in one hour on week day=(1/30)*60 =2

p(n=5 in one hour in week day) = e-10 105 /5! = 0.037

p(n=5 in one hour in weekend day) = e-2 25 /5! = 0.036

P(day is week day/n=5 in hour) =

p(day is week day)*p(n=5)/(p(day is week day)*p(n=5)+p(day is weekend)*p(n=5)

=(5/7)* 0.037/((5/7)* 0.037+(2/7)* 0.036)=0.837

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