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<m\\ (1-p)^{m-1}\sum_{k=m}^\infty t_k,& n= m.\end{cases} \end{aligned}

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<m\\ (p(1-p))^{m-1},& n= m.\end{cases}

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}

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