# Stopped Geometric Random Variable

A generalization of a problem I found on MSE: Suppose ${X\sim\mathrm{Geo}(p)}$ and ${T}$ is a nonnegative integer-valued random variable independent of ${X}$ with ${\mathbb E[T]<\infty}$. What is the distribution of ${Y:=X\wedge T}$?

For a fixed value of ${T}$, say ${T=m}$, we have

$\displaystyle \mathbb P(Y=n) = \mathbb P(X=n) = (1-p)^{n-1}p$

and so

$\displaystyle \mathbb P(Y=m) = 1 - \sum_{n=1}^{m-1}\mathbb P(Y=n) = (1-p)^{m-1}.$

Now let ${\mathbb P(T=m)=t_m}$. Then

\displaystyle \begin{aligned} \mathbb P(Y=n) &= \sum_{m=1}^\infty \mathbb P(Y = n\mid T = m)\mathbb P(T=m)\\ &= \sum_{m=1}^\infty \mathbb P(X\wedge m = n)t_m\\ &= \begin{cases} (1-p)^{n-1}p,& n

For an interesting example, suppose ${T\sim\mathrm{Geo}(1-p)}$. Then ${t_m = p^{m-1}(1-p)}$ so

$\displaystyle \sum_{k=m}^\infty t_k = \sum_{k=m}^\infty p^{k-1}(1-p) = p^{m-1}$

and

$\displaystyle \mathbb P(Y=n) = \begin{cases} (1-p)^{n-1}p,& n

Further,

\displaystyle \begin{aligned} \mathbb E[Y] &= 1 - (1-p)^{m-1} + m(p(1-p))^{m-1}\\ &= 1 - (1-p)^{m-1}(1-mp^{m-1}). \end{aligned}