Exercise 51

The joint density function for random variables X, Y, and Z is \(f(x, y, z) = Cxyz\) if \(0 \le x \le 2, 0 \le y \le 2, 0 \le z \le 2\), and \(f(x, y, z) = 0\) otherwise. (a) Find the value of the constant C. (b) Find \(P(X \le 1, Y \le 1, Z \le 1)\). (c) Find \(P(X + Y + Z \le 1)\).