If Y, is the total time between a customer's arrival in the store and departure from the servicewindow and if Y2 is the time spent in line before reaching the window, the joint density of thesevariables is0 < y2 < Y1 < ∞,elsewhere.se-Y1,f(V1. Y2) =The random variable Y, - Y2 represents the time spent at the service window.Find E (Y, – Y2). [Hint: Y, has a gamma distribution with a = 2 and ß = 1, and Y, has anexponential distribution with B = 1]%3D

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Answered: If Y, is the total time between a…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 32EQ

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For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)
Let X and Y be a pair of continuous random variables with a joint density fx,y(x,y). Assume that fx,y(x,y) = cxy for x greater than or equal to 0, y greater than or equal to 0, and x + y less than or equal to 1. Here c is a constant. Assume that fx,y(x,y) is 0 elsewhere. What is the constant c equal to? With the value of c, what is E[XY]?
Let X and Y be two random variables with joint density function f(x,y) = (3 − x + 2y) / 60, for 1 < x < 3, 0 < y < 5. Is P(X > 2, Y < 3) equal to P(X > 2) × P(Y < 3)?
Suppose that two continuous random variables X and Y have joint probability density function fxy = A( ex+y + e2x+y) , 1 ≤ x ≤ 2 ,0≤ y≤3 0 elsewhere a. P ( 3/2 ≤ X ≤ 2, 1 ≤ Y ≤ 2) b. Are the random variables X and Y independent? c. find the conditional density X given Y = 0
If the probability density of X is given by f(x) =kx3(1 + 2x)6 for x > 00 elsewhere where k is an appropriate constant, find the probabilitydensity of the random variable Y = 2X 1 + 2X . Identify thedistribution of Y, and thus determine the value of k.
If the independent random variables X and Y havethe marginal densitiesf(x) =⎧⎪⎪⎨⎪⎪⎩12for 0 < x < 20 elsewhereπ(y) =⎧⎪⎪⎨⎪⎪⎩13for 0 < y < 30 elsewherefind(a) the joint probability density of X and Y;(b) the value of P(X2 + Y2 > 1).
If the joint probability density of X and Y is given by f(x, y) =⎧⎪⎨⎪⎩13(x + y) for 0 < x < 1, 0 < y < 20 elsewherefind the variance of W = 3X + 4Y − 5.
Let x and y be joint continuous random variable with joint pdf f XY (x, y) = {cx+ 1, x, y ≥ 0, x + y < 10, otherwise1. Find the constant c.2. Find the marginal PDF’S fX (x) and fY (y)3. Find P(Y<2X2)
Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the temperature (◦F) at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint density
let X and Y be a random variables having pdf f(x,y)=2xy 0<x<y<1 Find P(X/Y<1/2)
A statistician has to decide on the basis of a singleobservation whether the parameter θ of the density f(x) =⎧⎨⎩2xθ 2 for 0 < x < θ0 elsewhere equals θ1 or θ2, where θ1 < θ2. If he decides on θ1 when theobserved value is less than the constant k, on θ2 when theobserved value is greater than or equal to the constant k,and he is fined C dollars for making the wrong decision,which value of k will minimize the maximum risk?

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