The number of emails Eva receives follows a Poisson process with rate X. Denote the number of emails she receives up to time t by N(t). Independent of everything else, the size of each email is an exponential random variable with mean 1/μ. a. Find E(N(5) - N(4) | N(4) = 3). b. Find pr(N(3) = 1 | N(1) = 1). c. Let X(t) represent the cumulative size of all emails that Eva has received by time t. Find E(X(t)). d. At t = 4, Eva has received a total of 3 emails. Denote S; as the arrival time of the ith email such that 0 ≤ S1 ≤ S2 ≤ S3 ≤ 4. Find E(S₁ + S2 + S3) and Pr(S₁ > 2).

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