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The number of buses that arrive at a bus stop during a one-hour time span can...

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Basic Math

The number of buses that arrive at a bus stop during a one-hourtime span can be modeled as a Poisson process with rate ? (seeRemark below). Now suppose a passenger has just arrived at the busstop and starts waiting. Let Y be the time (in unit of hours) sheneeds to wait to see the first bus.

(a) Is Y a discrete or continuous random variable? Find the setof all possible values of Y .

(b) For a possible value y of Y , find P(Y > y).

(c) Find the distribution (pdf/pmf) of Y and identify it as oneof the “named” distributions with corresponding parameter(s).

In the same setting as in the previous question, suppose we knowthere is exactly one bus arrival on a time interval [0, 1] (theunit is hour). Let Y be this single arrival time.

(d) Find the set of possible values of Y .

(e) For a possible value y of Y , find P(Y ? y).

(f) Find the pdf/pmf of Y and identify it as one of the “named”distributions with corresponding parameter(s).

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