Let the vector of random variables (X1, X2, X3) have the trinomial pdf with parameters n, p1, p2, and p3 = 1− p1 − p2. That is,
P(X1 = k1, X2 = k2, X3 = k3) = n!/k1! k2! k3! pk11 pk22 pk33,
ki = 0, 1, . . . , n; i = 1, 2, 3; k1 + k2 + k3 = n
By definition, the moment-generating function for (X1, X2, X3) is given by
MX1, X2, X3 (t1, t2, t3) = E(et1X1+t2 X2+t3 X3)
Show that
MX1,X2,X3 (t1, t2, t3)=(p1et1 + p2et2 + p3et3)n