Question

Exercises 169 Exercises 1 Let X and Y be continuous random variables having joint density function f. Find the joint distribution function and the joint density function of the random variables W-a+bX and Z = c+dY, where b > 0 and d0. Show that if X and Y are independent, then W and Z are independent. 2 Let X and Y be continuous random variables having joint distribution function F and joint density function f. Find the joint distribution function and joint density function of the random variables WX and Z = y2.

Answer 1

To find the joint distribution function and joint density function of the random variables W = a + bX and Z = c + dY, use the transformation method. If X and Y are independent, show that W and Z are independent by examining their joint density function.

Step-by-step explanation:

To find the joint distribution function and joint density function of the random variables W = a + bX and Z = c + dY, we need to use the transformation method. Let's start with the joint distribution function.

1. For W:

• Substitute W = a + bX into the original joint distribution function f(X, Y).
• Solve for X and differentiate with respect to X to find the marginal density function of X.

For Z:

• Substitute Z = c + dY into the original joint distribution function f(X, Y).
• Solve for Y and differentiate with respect to Y to find the marginal density function of Y.

Next, we can find the joint density function for W and Z by multiplying the marginal density functions of X and Y, giving us the product of the two.

To show that W and Z are independent if X and Y are independent, we can use the definition of independence and examine the joint density function of W and Z. If the joint density function can be expressed as the product of the marginal density functions of W and Z, then W and Z are independent.