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

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

(a) Is Y a discrete or continuous random variable? Find the set of 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 one of the “named” distributions with corresponding parameter(s).

In the same setting as in the previous question, suppose we know there is exactly one bus arrival on a time interval [0, 1] (the unit 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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milcah answered 1 week ago

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