help please?

Question

Let A, B, and C be r.v.'s. Below is their joint density function:

f_A,B,C of (a, b, c) =
2/3(a+b+c) when 0 ≤ a,b,c ≤ 1 and
0 elsewhere

find the probability that A ≤ 1/2.

Answer 1

In order to answer this, you need the density function for A, which you can obtain by integrating the joint density function with respect to b and c over all b and c.

It doesn't matter whether you integrate with respect to b or c first, and each is equally easy; so integrate first with respect to b, for b from 0 to 1 (keeping in mind that a and c should simply be treated as constants during the b-integration). You should get (2/3)(a + ½ + c). Next, integrate that result with respect to c, for c from 0 to 1. The result of that integration is the density function f_A(a). Then integrate that with respect to a for a from 0 to ½ to get the probability that A ≤ ½.

It wouldn't be a bad idea, as a check, to integrate f_A(a) with respect to a for a from 0 to 1 in order to confirm that you get 1.