Chapter 10.2, Problem 8Q

Let the vector of random variables $\left(X_1,X_2,X_3\right)$ have the trinomial pdf with parameters n $p_1,p_2,and{p}_3$. That is,

$P\left(X_1=k_1,X_2=k_2,X_3=k_3\right)=\frac{n!}{k_1!k_2!k_3!}{p}_1^{k_1}{p}_2^{k_2}{p}_3^{k_3},k_i=0,1,\mathrm{.....},n;\text{ i = 1,2,3; }{k}_1+k_2+k_3=n$

By definition, the moment-generating function for $\left(X_1,X_2,X_3\right)$ is given by

$M_{X_1,X_2,X_3}\left(t_1,t_2,t_3\right)=E\left(e^{t_1{X}_1+t_2{X}_2+t_3{X}_3}\right)$

Show that

$M_{X_1,X_1,X_1}(t_1,t_2,t_3)={\left(p_1{e}^{t_1}+p_2{e}^{t_2}+p_3{e}^{t_3}\right)}^n$