Question

Prove Corollary 5.9.2.

Short answer

The proof has been established.

Step-by-step solution

Given information

Corollary 5.9.2 states that Suppose that the random vector \(X = \left( {{X_1} \ldots {X_k}} \right)\) has the multinomial distribution with parameters \(n\) and \(p = \left( {{p_1} \ldots {p_k}} \right)\) with \(k > 2.\) Let \(l < k\), and let \({i_1} \ldots {i_l}\)be distinct elements of the set \(\left\{ {1, \ldots ,k} \right\}\)The distribution of \(Y = {X_{{i_1}}} + \ldots + {X_{{i_l}}}\)

is the binomial distribution with parameters \(n\)and \({p_{{i_1}}} + \ldots + {p_{{i_l}}}\)

Proof

Since \(X = \left( {{X_1} \ldots {X_k}} \right)\) it follows multinomial distribution, therefore the joint m.g.f. of \({X_1} \ldots {X_k}\)is

\(\begin{aligned}{}{M_{{X_1} \ldots {X_k}}}\left( {{t_1} \ldots {t_k}} \right) &= E\left( {{e^{{t_1}{X_1} + \ldots + {t_k}{X_k}}}} \right)\\ &= \sum\limits_x {\left[ {\frac{{n!}}{{{X_1}! \ldots {X_k}!}}p_1^{{x_1}} \ldots p_k^{{x_k}} \times {e^{{t_1}{X_1} + \ldots + {t_k}{X_k}}}} \right]} \\& = \sum\limits_x {\left[ {\frac{{n!}}{{{X_1}! \ldots {X_k}!}}{{\left( {{p_1}{e^{{t_1}}}} \right)}^{{x_1}}} \ldots {{\left( {{p_k}{e^{{t_k}}}} \right)}^{{x_k}}}} \right]} \\& = {\left( {{p_1}{e^{{t_1}}} + \ldots + {p_k}{e^{{t_k}}}} \right)^n}\end{aligned}\)

Therefore, the marginal distribution of \({X_i}\)is

\(\begin{aligned}{}{M_{{X_1}}}\left( t \right)& = {M_X}\left( {{t_1},0, \ldots 0} \right)\\ &= {\left( {{p_1}{e^{{t_1}}} + {p_2} + \ldots + {p_k}} \right)^n}\\ &= {\left[ {\left( {1 - {p_1}} \right) + {p_1}{e^{{t_1}}}} \right]^n}\\ \Rightarrow {X_1} \sim B\left( {n,{p_1}} \right)\\Similarly,\\ \Rightarrow {X_i} \sim B\left( {n,{p_i}} \right),\,\,i = 1,2 \ldots ,k\end{aligned}\)

It is given that \(k > 2\)and\(l < k\). Therefore, \(l = 1.\). This implies the parameter p is equal throughout all the individual binomial variates.

It is given that

\(Y = {X_{{i_1}}} + \ldots + {X_{{i_l}}}\)

\(\begin{aligned}{}{M_Y}\left( t \right)&= {M_{{X_{{i_1}}} + \ldots + {X_{{i_l}}}}}\left( t \right)\\ &= {M_{{X_{{i_1}}}}}\left( t \right) \ldots {M_{{X_{{i_l}}}}}\left( t \right)\\& = {\left( {q + p{e^t}} \right)^{{n_{_{{i_1}}}}}} \ldots {\left( {q + p{e^t}} \right)^{{n_{{i_l}}}}}\\& = {\left( {q + p{e^t}} \right)^n},\,\,where\,\,\sum\limits_{j = 1}^l {{n_{_{{i_j}}}} = n} \end{aligned}\)

We know that some of the Binomial variates are also a binomial variables with the same parameter \(p = {p_{{i_1}}} + \ldots + {p_{{i_l}}}\)

Therefore, \(Y \sim B\left( {n,p = {p_{{i_1}}} + \ldots + {p_{{i_l}}}} \right)\)