For $n=1,\cdots,$ let $X_n$ be the elapsed time between the occurrences of the $n-1$th and $n$th events of a Poisson Process with rate $\lambda$. Then $X_n$ is the exponential RV with parameter $\lambda$

Let $T$ be a RV taking positive values. Then $T$ has an exponential distribution if and only if it has the following memoryless property $$P(T>s+t|T>s)=P(T>t), \ \mbox{for all }s,t\ge0$$

Let $X_{1},X_{2},\ldots$ be a sequence of independent exponentially distributed random variables. Assume $X_{n}\sim\mbox{Exp}(\lambda_{n})$ for $\lambda_{n}>0$ and each integer $n\ge0$, then, 1. If $\sum\limits_{n=1}^{\infty} \frac{1}{\lambda_{n}}<\infty$, then $P\left(\sum\limits_{n=1}^{\infty}X_{n}<\infty\right)=1$ 2. If $\sum\limits_{n=1}^{\infty} \frac{1}{\lambda_{n}}=\infty$, then $P\left(\sum\limits_{n=1}^{\infty}X_{n}=\infty\right)=1$

Let $I$ be countable set. Let $\{T_{k}: k\in I\}$ be a sequence of independent exponentially distributed RVs such that $T_{k}\sim\mbox{Exp}(q_{k})$ and $0<q=\sum\limits_{k\in I}q_{k}<\infty$. Denote $T$ as the infimum of the infinite sequence: $$T=\inf\{T_{n}:n\in\mathbb{N} \}$$Then this infimum (or $T$) is attained at a unique random value $K$ of $k$ w.p.1 i.e. $$P(K<\infty)=1$$Moreover $T$ and $K$ are independent with $T\sim\mbox{Exp}(q)$ and $$P(K=k)=\frac{q_{k}}{q}$$ ## Definition from class (which kinda sucks) Now define $K$ as follows: if there exists unique $k\in I$ such that $T_{k}=T$, then $K=k$; otherwise $K=\infty$ i.e. $$K=\begin{cases}k&\exists\mbox{ unique }k\mbox{ s.t. }T_{k}=T \\ \infty&otherwise\end{cases}$$Then - $P(K<\infty)=1$ and $P(K=k)=\frac{q_{k}}{q}$ - $T\sim\mbox{Exp}(q)$ - $T$ and $K$ are independent