Suppose x, y, and z are random variables with the joint density function f(x, y, z) = ce−(0.5x + 0.2y + 0.1z) if x ≥ 0, y ≥ 0, z ≥ 0, and f(x, y, z) = 0 otherwise. (a) find the value of the constant c. (b) find p(x ≤ 1.375 , y ≤ 1.5). (round answer to five decimal places). (c) find p(x ≤ 1.375 , y ≤ 1.5 , z ≤ 1). (round answer to six decimal places).

a.

is a proper joint density function if, over its support, is non-negative and the integral of is 1. The first condition is easily met as long as . To meet the second condition, we require

b. Find the marginal joint density of and by integrating the joint density with respect to :

Then

c. This probability can be found by simply integrating the joint density:

b

step-by-step explanation: