Let random vector (X1, X2, X3) have joint pmf
f(x1, x2, x3) =
{
(1,0,0), (0,1,0), (0,0,1), (1,1,1) if (x1, x2, x3) = (11, 12, 13)
0 elsewhere
}
p(T1, T2, T3)
Denote M(t1, t2, t3) to be the mgf of (X1, X2, X3). Show that:
M(t1, t2, 0) = M(t1, 0, 0) * M(0, t2, 0)
M(t1, 0, t3) = M(t1, 0, 0) * M(0, 0, t3)
M(0, t2, t3) = M(0, t2, 0) * M(0, 0, t3)
But M(t1, t2, t3) ≠M(t1, 0, 0) * M(0, t2, 0) * M(0, 0, t3)
Thus, X1, X2, X3 are pairwise independent, but not mutually independent.
