Characteristic equation of binomial random variable

[sinned4789]

Homework Statement

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

Homework Equations

characteristic equation
I(t) = $$\sum$$p(x)*e$$^{tk}$$

The Attempt at a Solution

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

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

 

[LCKurtz]

In the definition

$$E(tX) = \sum_{k-0}^{n}\binom n k e^{tk}p^k(1-p)^{n-k}$$

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