Consider a Poisson process of intensity λ > 0 events per hour. Suppose that exactly one event occurs in the first hour. Let X be the time at which this event occurs, measured in hours (sothat the possible values of X are the reals in the interval [0, 1]).(a) What is P (0 ≤ X ≤ a) for a ∈ [0, 1]? (Of course, we are conditioning on exactly one event occurring in the first hour, as described.) Hint: Consider the number of events that occur in [0, a] and in (a, 1], similar to question #5 on the previous homework.(b) What is the distribution of X, under the same conditions? (You might want to wait until Monday’s class after break to do this last part.)

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Consider a Poisson process of intensity λ > 0 events per hour. Suppose that exactly one event occurs in the first hour. Let X be the time at which this event occurs, measured in hours (so

that the possible values of X are the reals in the interval [0, 1]).

(a) What is P (0 ≤ X ≤ a) for a ∈ [0, 1]? (Of course, we are conditioning on exactly one event occurring in the first hour, as described.) Hint: Consider the number of events that occur in [0, a] and in (a, 1], similar to question #5 on the previous homework.
(b) What is the distribution of X, under the same conditions? (You might want to wait until Monday’s class after break to do this last part.)

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