Exercise 52

Suppose X, Y, and Z are random variables with joint density function \(f(x, y, z) = Ce^{-(0.5x+0.2y+0.1z)}\) if \(x \ge 0, y \ge 0, z \ge 0\), and \(f(x, y, z) = 0\) otherwise. (a) Find the value of the constant C. (b) Find \(P(X \le 1, Y \le 1)\). (c) Find \(P(X \le 1, Y \le 1, Z \le 1)\).