Emails arrive in an inbox according to a Poisson process with rate (so the number of emails in a time interval of length t is distributed as Pois(t), 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.

(c) Find the conditional PMF of X + Y given that X + Y + Z = 36, and find E(X + Y | X + Y + Z = 36) and Var(X + Y | X + Y + Z = 36) (conditional expectation and conditional variance given an event are defined in the same way as expectation and variance, using the conditional distribution given the event in place of the unconditional distribution).

(b) Find the conditional joint PMF of X, Y,Z given that X + Y + Z = 36.

(c) Find the conditional PMF of X + Y given that X + Y + Z = 36, and find E(X + Y | X + Y + Z = 36) and Var(X + Y | X + Y + Z = 36) (conditional expectation and conditional variance given an event are defined in the same way as expectation and variance, using the conditional distribution given the event in place of the unconditional distribution).

Solution: (a) Since X ~ Pois(3lambda), Y ~ Pois(6lambda), Z ~ Pois(6lambda) independently, the joint PMF is the product of the marginals.
(b) (X, Y, Z) is conditionally Multinomial given T = t, and we have that (X, Y, Z) is conditionally Multinomial(36, (1/5, 2/5, 2/5)) given T = 36.
(c) Let W = X + Y and T = X + Y + Z. W|T = 36 ~ Bin (36, 9/15). Therefore, E(W|T = 36) = 21.6 and Var(W|T = 36) = 8.64.

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