How to compute the characteristic function

Question

Let $X_n$ be a sequence of identically distributed, independent random variables and let the distribution of $X_n$ be

$\mu_{X_n}(\{1\}) = a$ and $\mu_{X_n}(\{-1\}) = 1-a$ for 0 < a < 1.

Compute the characteristic function for the random variable $X_1 + X_2 + X_3 + ... + X_n$.

This is what i have so far. Is this correct?

$\phi(X_n) = E(e^{itXn}) = ae^{it} + (1-a)e^{-it}$

and since $X_n$ are independent the $ \Large \phi(\sum_{a=1}^{n} X_a) = E(e^{it\sum_{a=1}^{n} X_a}) = \prod_{a=1}^{n}E(e^{itX_a})$?

Answer 1

It seems you got the idea. Nevertheless:

• since $a$ is a parameter, you should not use it as an index in the sum and the product.
• As pointed out by William, you can simplify the expression and get $$\varphi_{X_1+\dots+X_n}(t)=\left(ae^{it}+(1-a)e^{-it}\right)^n.$$