Question

Random variables X and Y have the joint PMF pₓ,ᵧ(x,y)={k(x²+y²)

0 if x∈{0,1,2},y∈{2,3,4,5}
What is the value of the constant k ?

Answer 1

To determine the value of k, we sum the values of k(x² + y²) for all x and y in their respective sets, set this sum equal to 1, and solve for k, ensuring the total probability sums to 1 as required for a PMF.

Step-by-step explanation:

The student has provided the joint probability mass function (PMF) for random variables X and Y, and is seeking to find the value of the constant k. The values that X and Y can take are given, and the constraints of the PMF are such that:

• The sum of all the probabilities must equal 1.
• Each probability value p₁ₓᵢ(x, y) is greater than or equal to 0 and less than or equal to 1.

Therefore, to find k, we sum all the possible values of p₁ₓᵢ(x, y) for x in {0, 1, 2} and y in {2, 3, 4, 5} and set the sum equal to 1. Then, we solve for k. Let's set up and solve the equation:

1. Calculate the sum of k(x² + y²) over all x and y values in their given sets.
2. Set the sum equal to 1 because the total probability must equal 1.
3. Solve for the constant k.

Once we find k, the PMF will be properly scaled to reflect the probabilities of X and Y.