   Chapter 15.6, Problem 51E

Chapter
Section
Textbook Problem

The joint density function for random variables X, Y, and Z is f(x, y, z) = Cxyz if 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, 0 ≤ z ≤ 2, and f (x, y, z) = 0 otherwise.(a) Find the value of the constant C.(b) Find P(X ≤ l, Y ≤ 1,Z ≤ 1).(c) Find P (X + Y + 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)={Cxyz , if 0x2,0y2,0z2  0   ,  otherwise

Calculation:

3f(x,y,z)dA=f(x,y,z)dA=000f(x,y,z)dzdydx+020202f(x,y,z)dzdydx+222f(x,y,z)dzdydx=0+020202f(x,y,z)dzdyd

(b)

To determine

To find: The value of P(X1,Y1,Z1).

(c)

To determine

To find: The value of P(X+Y+Z1).

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