Question

The joint probability density function of two continuous random variables X and Y is given by,

f ( x, y ) = { C (e^ (− y))/ x^ 2 1 < x < [infinity] , 1 < y < [infinity]

0 otherwise

(a) Compute the value of C .

(b) Determine if X is independent of Y by computing f XIY ( x | y ) and comparing it to f X ( x ) (alternatively, you may compare f Y IX ( y | x ) to f Y ( y )).

Answer 1

To determine the constant C in the joint pdf, we integrate the function over its support and set the result equal to 1. For independence between X and Y, we compare the product of the marginal densities to the joint pdf.

Step-by-step explanation:

Calculating the Constant C in the Joint Probability Density Function

To find the value of C, we must ensure that the total probability for the joint probability density function equals 1. This is given by the integral over the entire support of X and Y:

\[1 = \int_{1}^{\infty}\int_{1}^{\infty} C \frac{e^{-y}}{x²} dy dx\]

Solving this double integral leads to the value of C.

Checking for Independence between X and Y

To check if X and Y are independent, we need to find the marginal density functions fX(x) and fY(y) by integrating the joint pdf over the possible values of the other variable. If the product of the two marginals, fX(x) * fY(y), equals the joint pdf f(x, y), then X and Y are independent.