PMF of function of random variables

Question

Let X be a Geom($\frac{1}{2}$) random variable, and define Y=$X^{-1}$

What is the p.m.f. of Y ?

attempt:

pmf of a Geom RV in general form is $p(1-p)^{k-1}$

There is this similar question, not for the same type of RV, but it doesn't actually explain, the accepted answer states "You can finish it from there"

I've also watched this video, but in the example the range of values is explicitly defined.

I'm not sure how to relate the p.m.f of X to p.m.f of Y for this specific case of mine

Answer 1

Since $X$ is concentrated on the positive integers, $Y=1/X$ is concentrated on the set $S=\{1/k\colon\, k\geq 1\}$ i.e. $P(Y\in S)=1$. Hence it sufficies to compute the probability mass function of $Y$ for points in $S$. But $$ P(Y=k^{-1})=P(X=k)=2^{-k} $$ for $k=1,2,3, \dotsc$ as desired.

Answer 2

$X$ takes positive integer valyes and $Y$ takes the values $1,\frac 1 2,\frac 1 3,...$. We have $P(Y=\frac 1 n)=P(X=n)=p(1-p)^{n-1}$.