prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$
How to prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$.
By memoryless property,
$$P(X=i+s | X>i)=P(X=s)$$
How to get distribution for X from the above ?
Let $p=P[X\geqslant2]$, then $P[X\geqslant i+1\mid X\geqslant i]=p$ for every $i\geqslant1$ hence $P[X\geqslant i]=p^{i-1}$ for every $i\geqslant1$. Surely you can deduce the distribution of $X$ from this observation.