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In: Statistics and Probability

Emails arrive in an inbox according to a Poisson process with rate λ (so the number...

Emails arrive in an inbox according to a Poisson process with rate λ (so the number of emails in a time interval of length t is distributed as Pois(λt), and the numbers of emails arriving in disjoint time intervals are independent). Let X, Y, Z be the numbers of emails that arrive from 9 am to noon, noon to 6 pm, and 6 pm to midnight (respectively) on a certain day. (a) Find the joint PMF of X, Y, Z. (b) Find the conditional joint PMF of X, Y, Z given that X + Y + Z = 36.

Solutions

Expert Solution

The probability mass function of X, the number of emails that arrive from 9 AM to Noon i.e., for a time interval of length 3 (=12 – 9) hrs is distributed as Pois(3λ),

   … x = 0,1,2,3,…

The probability mass function of Y, the number of emails that arrive from Noon to 6 PM i.e., for a time interval of length 6 (=12 – 6) hrs is distributed as Pois(6λ),

… y = 0,1,2,3,…

The probability mass function of Z, the number of emails that arrive from 6 PM to Midnight i.e., for a time interval of length 6 (=12 – 6) hrs is distributed as Pois(6λ),

… z = 0,1,2,3,…

(a)

Since the number of emails arriving in disjoint time intervals are independent, the joint probability mass function (PMF) of X, Y, and Z is

(b)

Since, Sum of independent Poisson random variables are Poisson random variables. Hence

… w = 0,1,2,3,…

The conditional joint PMF of X, Y, Z given that X + Y + Z = 36

orchestra answered 2 years ago
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