Assume that the Poisson process X = {X(t) : t ≥ 0} describes students’ arrivals at...

Assume that the Poisson process X = {X(t) : t ≥ 0} describes students’ arrivals at the library with intensity λ = 4 per hour. Given that the tenth student arrived exactly at the end of fourth hour, or W₁₀ = 4, find:

1. E [W₁|W₁₀ = 4]

2. E [W₉ − W₁|W₁₀ = 4].

Hint: Suppose that X {X(t) : t ≥ 0} is a Poisson process with rate λ > 0 and its arrival times are defined for any natural k as W_k = min[t ≥ 0 : X(t) = k] (1) Then for any natural m, the inter-arrival times, {T₁ = W₁, T₂ = W₂ − W₁, . . . , T_m = W_m − W_m−1} are independent variables with the common exponential distribution, f^T(t) = λ · e −λ·t for t > 0.

Expert Solution

orchestra answered 1 year ago

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