The joint density function for random variables x, y, and z is f(x, y, z)= cxyz if 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ 2, and f(x, y, z) = 0 otherwise. (a) find the value of the constant c. (b) find p(x ≤ 1, y ≤ 1, z ≤ 1). (c) find p(x + y + z ≤ 1).

In order for to be a proper density function, we need to have

where is the support of the joint distribution. Here is the box with dimensions 1x2x2. We have

Now since we can write as

(where ) we can assign each factor to be the PDF for the corresponding random variable, i.e.

which means that are independent of one another. Then

We can derive the marginal distributions in the same way we solve for above:

with defined similarly, so that

For the final probability, we condition on the event that and :

The CDF of is easy to find:

So we have