   Chapter 15.4, Problem 28E

Chapter
Section
Textbook Problem

(a) Verify that f ( x , y ) = { 4 x y if   0   ≤   x   ≤   1 ,       0   ≤   y   1 0 otherwise is a joint density function.(b) If X and Y are random variables whose joint density function is the function f in part (a), find(i) P (X ≥ 1 2 ) (ii) P(X ≥ 1 2 , Y ≤ 1 2 )(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)={4xy , if 0x1,0y1  0   ,  otherwise .

The region D is  0x1,0y1 .

Calculation:

Obtain 2f(x,y)dA .

2f(x,y)dA=f(x,y)dA=00f(x,y)dydx+0101f(x,y)dydx+11f(x,y)dydx=0+∫<

(b) (i)

To determine

To find: The value of P(X12) .

(ii)

To determine

To find: The value of P(X12,Y12) .

(c)

To determine

To find: The expected value of X and Y.

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