how to determine the characteristic function

Question

Problem 2.24 from Digital Communications, 5th Edition by John Proakis, Masoud Salehi

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$Y=\sum_{i=1}^nXi$

where the $Xi, i = 1, 2, . . . , n, $ are statistically independent random variables with

$Xi = \begin{cases} 1\ with\ probability\ p \\ 0\ with\ probability\ 1−p \\ \end{cases}$

Determine the characteristic function of Y

thank you .

Answer 1

First identify that $X$ is Bernoulli distributed.

So we have $P(X=1) = p = 1 - P(X=0)$, and by using the formula for $E[h(x)]$

you get that $E[e^{itX}] = \sum e^{ikt} P(X=k) = P(X=0) e^{0} + P(X=1)e^{it} = 1-p + pe^{it}$.

Can you take it from here?