The question:
Let $X_{1},X_{2}, ... ,X_{k-1}$ have a multinomial distribution.
(a)Find the mgf of $X_{2},X_{3}, ... ,X_{k-1}$.
(b)What is the pmf of $X_{2},X_{3}, ... ,X_{k-1}$?
(c)Determine the conditional pmf of $X_{1}$ given that $X_{2}=x_{2}, ...,X_{k-1}=x_{k-1}.$
(d)What is the conditional expectation $E(X_{1}|\ x_{2},... ,x_{k-1})$?
My attempt:
(a) The mgf of a multinomial distribution is given by $M(t_{1}, ... ,t_{k-1})=(p_{1}e^{t_1}+...+p_{k-1}e^{t_{k-1}}+p_{k})^n$ for all real values $t_1,t_2, ... ,t_{k-1}$. So,$E(e^{t_{2}X_{2}+ ... +t_{k-1}X_{k-1}})=M(0,t_{2},... ,t_{k-1})=(1+p_{2}e^{t2}+...+p_{k-1}e^{t_{k-1}}+p_{k})^n$.
(b)The pmf of $X_{2},X_{3}, ... ,X_{k-1}$ is
$\sum_{x_1=0}^{n}\frac{n!}{x_1!x_2! \cdots x_{k-1}!(n-x_1-x_2- \cdots x_{k-1})!}p_{1}^{x_1}p_{2}^{x_2}\cdots p_{k}^{n-x_1-x_2-\cdots-x_{k-1}}$. I'm stuck here in calculating the summation.
(c),(d) I wonder whether the pmf of $X_{1}$ given that $X_{2}=x_{2}, ...,X_{k-1}=x_{k-1}$ is
Binomial$(n-x_2-\cdots-x_{k-1},\frac{p_1}{1-p_2-\cdots p_{k-1}})$.