Show that a memory-free random variable has a Geometric distribution

Question

Let X be a discrete random variable that takes values in $\Bbb{N}$ . Show that if X is memory-free, then X∼ Geom (p) for some p. So far I'm trying to show that $Pr(X>t) = Pr(X>1)^t$ for all t ∈ $\Bbb{N}$, but I will appreciate some additional help.

Answer 1

By memeoryless on $\mathbb N$, I assume you mean something like $$\mathbb{P}(X=t+s | X\gt t) = \mathbb{P}(X=s)$$ for $s,t \in \mathbb{N}$ perhaps with some adjustment if you want to include $0$ in $\mathbb{N}$

Let's define $p=\mathbb{P}(X=1)$ and let $s=1$ so as to get $\mathbb{P}(X=t+1 | X\gt t) = p$

It is then a simple induction that $\mathbb{P}(X=t)= p(1-p)^{t-1}$ and $\mathbb{P}(X \gt t)= (1-p)^t = \mathbb{P}(X \gt 1)^t$

• clearly true for $t=1$ as $\mathbb{P}(X=1)= p$ and $\mathbb{P}(X \gt 1) = 1-p$
• if true for $t=n$, then $\mathbb{P}(X=n+1) =\mathbb{P}(X=n+1 | X\gt n) P(X \gt n) = p(1-p)^n$ so $\mathbb{P}(X \gt n+1) = \mathbb{P}(X \gt n) - \mathbb{P}(X=n+1) = (1-p)^n- p(1-p)^n = (1-p)^{n+1}$, i.e. true for $t=n+1$