Choose an even integer randomly from the set 0 2 4 6 8 Then

Choose an even integer randomly from the set {0, 2, 4, 6, 8}. Then choose an integer randomly from the set {0, 1, 2, 3, 4}. Assume X equal the integer that is selected from the first set and let Y equal the sum of the integers. Find that are X and Y independent ? Why or Why not?

Solution

Two random variables X and Y are independent if and only if f1(x)f2(y)=f(x,y). Since f(0,0)=1/25, we have f1(0)f2(0)!=f(0,0). Thus, we can conclude that X and Y are not independent, that is they are dependent. Moreover note that the support is not rectangular, and we know that when the support is not rectangular random variables must ne dependent.