   Chapter 15.4, Problem 29E

Chapter
Section
Textbook Problem

Suppose X and Y are random variables with joint density function f ( x , y ) = { 0 0.1   e − ( 0.5 x + 0.2 y )   o t h e r w i s e i f   x ≥ 0 ,   y ≥ 0   (a) Verify that f is indeed a joint density function.(b) Find the following probabilities.(i) P(Y ≥ 1)(ii) P(X ≤ 2, Y ≤ 4)(c) Find the expected values of X and Y.

(a)

To determine

To verify: The given function is the joint density function.

Explanation

Property used:

If the given function f(x,y) is the non-negative, then it is called the joint density function if it satisfies the equation 2f(x,y)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)dydx=abg(x)dxcdh(y)dy (1)

Given:

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

Calculation:

Obtain 2f(x,y)dA .

2f(x,y)dA=f(x,y)dA=00f(x,y)dydx+00f(x,y)dydx=0+00f(x,y)dydx=00f(x,y)dydx

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

00f(x,y)dydx=000.1e(0.5x+0.2y)dydx=0.100e0.5xe0.2ydydx=0.10e0.5xdx0e0

(b) (i)

To determine

To find: The value of P(Y1) .

(ii)

To determine

To find: The value of P(X2,Y4) .

(c)

To determine

To find: The expected value of X and Y.

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