Question

Determine the values of c so that the following functions represent joint probability distributions of the random variables X and Y : f(x, y) = c|x − y|, for x = −2, 0, 2; y = −2, 3.

Answer 1

f(-2, -2) = c|-2-(-2)| = 0

f(-2, 3) = c|-2-3| = 5c

f(0, -2) = c|0-(-2)| = 2c

f(0, 3) = c|0-3| = 3c

f(2, -2) = c|2-(-2)| = 4c

f(2, 3) = c|2-3| = c

The total probability is a sum of these probabilities:

1 = f(-2,-2) + f(-2,3) + f(0,-2) + f(0, 3) + f(2, -2) + f(2,3) = 0 +5c + 2c + 3c + 4c + c = 15c.

Therefore, c = 1/15.

Answer. c =1/15