Question

Suppose that X and Y are random variables with joint probability density function

x²-1
fₓ,ᵧ (x,y)= ____, if 1 ≤ x ≤ 2and 0 ≤ y ≤ 1,
k
0, otherwise

​Find k so that fₓ,ᵧ (x,y) is a valid joint probability density function.

Answer 1

To find the value of k in the joint probability density function x²-1fₓ,ᵧ(x,y), we integrate the function over its entire range and set it equal to 1.

Step-by-step explanation:

The joint probability density function (PDF) needs to satisfy the condition that the integral over its entire range is equal to 1. In this case, we have the joint PDF x²-1fₓ,ᵧ(x,y) given as k, if 1 ≤ x ≤ 2 and 0 ≤ y ≤ 1, and 0 otherwise. To find the value of k, we need to integrate the joint PDF over its entire range and set it equal to 1:

∫∫ (x²-1)k dxdy = 1

Integrating the joint PDF with respect to x first, we get:

k(∫1²-1)dy = 1

Integrating both sides with respect to y:

k(1-0) = 1

k = 1

Therefore, the value of k that makes the joint PDF a valid probability density function is k = 1.