Question 3 of 43. The arrival time of a bus, measured exactly, has a discrete sample space.
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A man arrives at a bus stop at a random time (that is, with no regard for the scheduled service) to catch the next bus. Buses run every 40 minutes without fail, hence the next bus will come at any time during a 40-minute interval with a uniformly distributed probability. Let X be the number of minutes before the next bus arrives at the stop: (a) Find P(X < 12). (b) Find the probability that the bus will come after the next 20 minutes but within 10 minutes.
Adi S.
You arrive at a bus stop at 10 clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. 1. (3 points) What’s the probability that you will have to wait longer than 10 minutes. 2. (3 points) If, at 10:10, the bus has not arrived yet. What’s the probability that you will have to wait at least an additional 5 minutes?
1.9. Arrivals of passengers at a bus stop form a Poisson process X(t) with a rate of 2 per unit time. Assume that a bus departed at time t, leaving no customers behind. Let T denote the arrival time of the next bus. Then the number of passengers present when it arrives is X(T). Suppose that the bus arrival time T is independent of the Poisson process and that T has the uniform probability density function for 0 ≤ t ≤ 1. (a) Determine the conditional moments E[X(T)|T = t] and E[{X(T)}²|T = t]. (b) Determine the mean E[X(T)] and variance Var[X(T)].
Sri K.
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