how to prove binomial through bernoulli indicators ??

Question

how to prove binomial through Bernoulli indicators? is it x-bernoulli (P) y=x1,x2,...,xn. where xi is the independent variable bernoulli gives y-Bin(n,p)?

Answer 1

let X1,X2,...,Xn are iid Bernoulli random variables with parameter p then Xi has probability distribution:

f(xi;p)=p^xi * (1-p)^(1-xi) ; xi=0 or 1

& the moment generating function of Xi is

M_Xi(t)=(1-p)+pe^t

let Y=X1+X2+...+Xn

the moment generating function of Y is

M_Y(t)=M_X1+X2+...+Xn(t)=M_X1(t) * M_X2(t)*...*M_Xn(t)

                    ={(1-p)+pe^t}*{(1-p)+pe^t}*...*{(1-p)+pe^t}

                    ={(1-p)+pe^t}^n

which is the MGF of Binomial distribution

so y-Bin(n,p)