Let $X$ be a discrete random variable taking only non-negative integer values . Let $f(s):=\sum_{n=0}^\infty P(X=n)s^n$
(think of the sum formally, and indeed it absolutely converges for $|s|<1)$ .
Then how do we write $g(s)=\sum_{n=0}^\infty P(X=2n)s^n$ in terms of $f$ ?
I know that $\sum _{n=0}^\infty P(X>n)s^n=\dfrac {1-f(s)}{1-s}$ for $|s|<1$, but I don't know whether it is useful or not .
Please help . Thanks in advance
You cannot write $g(\cdot)$ in terms of $f(\cdot)$ alone unless you know more about the distribution of $X$.
However $\lower{3.5ex}{\begin{split}g(s^{2}) &= \sum_{n=0}^\infty \mathsf P(X=2n) s^{2n} \\ &= f(s)-\sum_{n=0}^\infty \mathsf P(X=2n+1)s^{2n+1}\end{split}}$