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Question 1: The management of a fast-food outlet is interested in the joint behavior of the random variables Y1 = total time between a customer's arrival at the store and departure from the service window; and Y2 the time 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 bivariate density function for Y1 and Y2 is given by:
e-y1, 0 ≤ y2 ≤ y1 < ∞
f(Y1, Y2)= {
0, Otherwise
with time measured in minutes.
a. Find P(Y1 < 2, Y2 > 1).
b. Find P(Y1 ≥ 2Y2)
c. Find P(Y1 ≥ Y2 ≤ 1)
Question 2:
Consider two random variables Y1 and Y2 with the density function
2(1- y1), 0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1
f(Y1, Y2) = {
0, otherwise.
Derive the density function of the random variable Z = Y1Y2
Question 3:
Let X1, X2, ...Xn be n random variables which are pairwise independent. Then
V(Σi=1nXi) = Σi=1nV(Xi).
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