Help me pleaseeeee !!!!

Consider a discrete random vector x = (X₁,..., XK) that follows the multinomial distribution with the
following joint mass function.
fa (x1,...,xK; n, P₁,..., PK)
1
Show that the moment generating function of x is as follows.
Me (t1,..., tk; n, P₁,...,1
K
Σ ak
k=1
Recall that the support of this distribution is the set of points (x₁,...,xK) E NK such that
x = {0, 1,...,n} for k = 1,..., K and EK_₁ *k = n.
n
=
n!
PK) =
k=1
where X is a set identical to the support of x.
K
; ПРЕ
k=1
To complete this derivation you should take advantage of the multinomial theorem, which
states that the n-th power of a sum of K values a₁,..., ak is calculated as:
*k!
n!
Σ
(*1,...,.*K) EX K-1 k!
K
- ΣPk exp (tk)
k=1
(tu))"
K
II ak
k=1
Calculate the following three moments:
• the mean E[X] of the k-th random variable listed in x
• the variance Var [X] of the k-th random variable listed in x
• Give an intuitive interpretation for the expression of all the three moments above.