On average I receive 40 emails a day Find the probability th

On average I receive 40 emails a day. Find the probability that
(a) I get exactly 45 emails tomorrow
(b) I get exactly 2 emails in the next hour
(c) I get more than 2 emails in the next hour
(d) The next email doesn

Solution

This is poission distribution,

mean = m = 40 mails

P(x=r) = e^(-m) * m^r / r!

P(x=45) = e^(-40) * 40^45 /45! = 0.043965

Now,

b.

mean for one hour = mh = 24/40 = 5/3

P(x=2) = e^(-5/3) * (5/3)^2/2! = 0.2623

c.

P(x>2) = 1 - [ P(x=0) + P(x=1) + P(x=2) ]

P(x>2) = 1 - [ 0.1888 + 0.3147 + 0.2623 ]

P(x>2) = 1 - 0.7659

P(x>2) = 0.2341

d.

mean for 15 minutes,

m15 = 40 / 96 = 5/12 = 0.4167

no email in 15 minute,

P(x=0) = e^(-0.4167) * (0.4167)^0 / 0! = 0.6592