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Suppose that Y1 is the total time between a customer’s arrival in the store and departure from the service window, Y2 is the time spent in line before reaching the window, and the joint density of these variables (as was given in Exercise 5.15) is
- a Find the marginal density
functions for Y1 and Y2. - b What is the conditional density function of Y1 given that Y2 = y2? Be sure to specify the values of y2 for which this conditional density is defined.
- c What is the conditional density function of Y2 given that Y1 = y1? Be sure to specify the values of y1 for which this conditional density is defined.
- d Is the conditional density function f(y1|y2) that you obtained in part (b) the same as the marginal density function f1(y1) found in part (a)?
- e What does your answer to part (d) imply about marginal and conditional probabilities that Y1 falls in any interval?
5.15 The management at a fast-food outlet is interested in the joint behavior of the random variables Y1, defined as the total lime between a customer’s arrival at the store and departure from the service window, and Y2, the lime a customer waits in line before reaching the service window. Because Y1 includes the time a customer waits in line, we must have Y1 ≥ Y2. The relative frequency distribution of observed values of Y1 and Y2 can be modeled by the
with time measured in minutes. Find
- a P(Y1 < 2, Y2 > 1).
- b P(Y1 ≥ 2Y2).
- c P(Y1 – Y2 ≥ 1). (Notice that Y1 – Y2 denotes the time spent at the service window.)
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Chapter 5 Solutions
Mathematical Statistics with Applications
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