   Chapter 15.6, Problem 52E

Chapter
Section
Textbook Problem

Suppose X, Y, and Z are random variables with joint density function f ( x ,   y   ,   z )   =   C e − ( 0.5 x   +   0.2 y   +   0.1 z ) i f   x   ≥   0 ,   y   ≥   0 ,   z ≤   0 ,   and f (x, y, z) = 0 otherwise.(a) Find the value of the constant C.(b) Find P(X ≤ 1, Y ≤ 1).(c) Find P(X ≤ 1, Y ≤ 1, Z ≤ 1).

(a)

To determine

To find: The value of a constant C.

Explanation

Property used:

If the given function f(x,y,z) is the joint density function then, it satisfies the equation 3f(x,y,z)dA=1.

Formula used:

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)k(z)dzdydx=abg(x)dxcdh(y)dyefk(z)dz (1)

Given:

The joint density function, f(x,y,z)={Ce(0.5x+0.2y+0.1z) , if x0,y0,z0,  0   ,  otherwise

Calculation:

3f(x,y,z)dA=f(x,y,z)dA=000f(x,y,z)dzdydx+000f(x,y,z)dzdydx=0+000f(x,y,z)dzdydx=000f(x,y,z)dzdydx

Find the value of the triple integral using the equation (1),

000f(x,y,z)dzdydx=000Ce(0.5x+0.2y+0.1z)dzdydx=000C(e0

(b)

To determine

To find: The value of P(X1,Y1).

(c)

To determine

To find: The value of P(X1,Y1,z1).

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Convert the expressions in Exercises 31-36 to positive exponent form. 45y3/4

Finite Mathematics and Applied Calculus (MindTap Course List)

In Exercises 1124, find the indicated limits, if they exist. 17. limx32x35

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Find the derivatives of the functions in Problems 1-10. 9.

Mathematical Applications for the Management, Life, and Social Sciences

True or False: A trigonometric substitution is necessary to evaluate xx2+10dx.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Find for y defined implicity by .

Study Guide for Stewart's Multivariable Calculus, 8th

Find each value of x. logx33=12

College Algebra (MindTap Course List) 