I am trying to write the following expression mathematically: let $X_1,X_2$ be independent random variables, each having the probability function $P_X(k)=p(1-p)^k$ for k=0,1,... and $0<p<1$.
Is it something like $\forall X_1X_2 \exists p_X(k)=p(1-p)^k|(k=0,1,2,..) \land (0<p<1)$ ?
I would appreciate a correction and explanation, so I could not repeat the same mistake in the future.
Note: I did not define what is an independent random variable there; I just gave a general expression for random variables $X_1,X_2$. If you could, I would appreciate seeing how to write it with the definition of independence.
Thank you very much!