Help with parts c thru f 1.A bank operates both a drive-up facility and a walk-up window. On a randomly selectedday, let X be the proportion of time that the drive-up facility is in use (at least one customer is beingserved or waiting to be served) and Y be the proportion of time that the walk-up window is in use. Thejoint pdf of (X, Y) is given by(a)(b)(c)(d)(e)f(x, y)=(x+y²) 0≤x≤ 1,0 ≤ y ≤ 1otherwiseFind the probability that neither facility is busy more than one-quarter of the time.Find the marginal densities of X and Y.Using the marginal density of Y from (b), find the probability that the walk-up windowis in use between 1/4th and 3/4 of the time.1Are X and Y independent?Find the conditional density of Y given that the drive-up facility is in use 80% of thetime, i.e. X = 0.8.(f)Use the conditional density found in part (e) to find that the walk-up facility is busy atmost half the time given that X = 0.8..(Hint: Integrate the density obtained in (e) over the required bounds)

(d) (e) Using the marginal density of Y from (b), find the probability that the walk-up window is in use between 1/4th and 3/4 of the time. 1 Are X and Y independent? Find the conditional density of Y given that the drive-up facility is in use 80% of the time, i.e. X = 0.8. (f) Use the conditional density found in part (e) to find that the walk-up facility is busy at most half the time given that X = 0.8.. (Hint: Integrate the density obtained in (e) over the required bounds)

(d) (e) Using the marginal density of Y from (b), find the probability that the walk-up window is in use between 1/4th and 3/4 of the time. 1 Are X and Y independent? Find the conditional density of Y given that the drive-up facility is in use 80% of the time, i.e. X = 0.8. (f) Use the conditional density found in part (e) to find that the walk-up facility is busy at most half the time given that X = 0.8.. (Hint: Integrate the density obtained in (e) over the required bounds)

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

Help with parts c thru f

1.
A bank operates both a drive-up facility and a walk-up window. On a randomly selected
day, let X be 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 be the proportion of time that the walk-up window is in use. The
joint pdf of (X, Y) is given by
(a)
(b)
(c)
(d)
(e)
f(x, y)
=
(x+y²) 0≤x≤ 1,0 ≤ y ≤ 1
otherwise
Find the probability that neither facility is busy more than one-quarter of the time.
Find the marginal densities of X and Y.
Using the marginal density of Y from (b), find the probability that the walk-up window
is in use between 1/4th and 3/4 of the time.
1
Are X and Y independent?
Find the conditional density of Y given that the drive-up facility is in use 80% of the
time, i.e. X = 0.8.
(f)
Use the conditional density found in part (e) to find that the walk-up facility is busy at
most half the time given that X = 0.8..
(Hint: Integrate the density obtained in (e) over the required bounds)

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