2. A bank operates both a drive-up facility and a walk-up window. On a randomly selected day, let X = the proportion of time that the drive-up facility is in use (at least one customer is being served or waiting to be served) and Y = the proportion of the time that the walk-up window is in use. Then the set of possible values for (X, Y) is the rectangle {(x, y): 0 ≤ x ≤ 1, 0≤ y ≤ 1}. Suppose the joint probability density function of (X, Y) is given by: (x + y²) 0≤x≤ 1,0 ≤ y ≤ 1 fxx(x, y) = {(x + y²) 0 Otherwise (a) Show that fx,y is legitimate (b) Find the probability that neither facility is busy more than one-quarter of the time. (c) Find fx and fy, the marginal probability density function of X and Y respectively. (d) Construct the conditional probability density function of Y given that X=0.8. (e) Evaluate the probability that the walk-up facility is busy at most half of the time given that X=0.8 (f) Calculate the expected proportion of the time that the walk-up facility is busy given that X=0.8 Tutor: Takembo Clovis (L)
2. A bank operates both a drive-up facility and a walk-up window. On a randomly selected day, let X = the proportion of time that the drive-up facility is in use (at least one customer is being served or waiting to be served) and Y = the proportion of the time that the walk-up window is in use. Then the set of possible values for (X, Y) is the rectangle {(x, y): 0 ≤ x ≤ 1, 0≤ y ≤ 1}. Suppose the joint probability density function of (X, Y) is given by: (x + y²) 0≤x≤ 1,0 ≤ y ≤ 1 fxx(x, y) = {(x + y²) 0 Otherwise (a) Show that fx,y is legitimate (b) Find the probability that neither facility is busy more than one-quarter of the time. (c) Find fx and fy, the marginal probability density function of X and Y respectively. (d) Construct the conditional probability density function of Y given that X=0.8. (e) Evaluate the probability that the walk-up facility is busy at most half of the time given that X=0.8 (f) Calculate the expected proportion of the time that the walk-up facility is busy given that X=0.8 Tutor: Takembo Clovis (L)
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 28EQ
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Transcribed Image Text:2. A bank operates both a drive-up facility and a walk-up window. On a
randomly selected day, let X = the proportion of time that the drive-up facility
is in use (at least one customer is being served or waiting to be served) and Y
= the proportion of the time that the walk-up window is in use. Then the set
of possible values for (X, Y) is the rectangle
D = {(x, y): 0 ≤ x ≤ 1,0 ≤ y ≤ 1}. Suppose the joint probability density
function of (X, Y) is given by:
fx,y(x,y) = { / (x + y ² )
²²(x + y²) 0≤x≤ 1,0 ≤ y ≤ 1
Otherwise
0
(a) Show that fx,y is legitimate
(b) Find the probability that neither facility is busy more than one-quarter of
the time.
(c) Find fx and fy, the marginal probability density function of X and Y
respectively.
(d) Construct the conditional probability density function of Y given that
X=0.8.
(e) Evaluate the probability that the walk-up facility is busy at most half of the
time given that X=0.8
(f) Calculate the expected proportion of the time that the walk-up facility is
busy given that X=0.8
Tutor: Takembo Clovis (L)
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