Characteristic equation of binomial random variable

[sinned4789]

Homework Statement

find the characteristic equation of a binomial variable with pmf p(x) =[tex]\frac{n!}{(n-k)!k!}[/tex]*p[tex]^{k}[/tex]*(1-p)[tex]^{n-k}[/tex]

Homework Equations

characteristic equation
I(t) = [tex]\sum[/tex]p(x)*e[tex]^{tk}[/tex]

The Attempt at a Solution

I(t) = [tex]\sum[/tex][tex]\frac{n!}{(n-k)!k!}[/tex]*(p[tex]^{k}[/tex]*(1-p)[tex]^{-k}[/tex]*e[tex]^{tk}[/tex])*(1-p)[tex]^{n}[/tex]

i am stuck on this series because i don't know what to do with the combination term.

 

[LCKurtz]

In the definition

[tex]E(tX) = \sum_{k-0}^{n}\binom n k e^{tk}p^k(1-p)^{n-k}[/tex]

group the e^tk with the p^k and then think about what the binomial expansion of (a + b)ⁿ looks like.