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I want to understand the Poisson and exponential distributions correctly. Would this be correct "If $X$ follows a Poisson distribution, then $T$ measures the probability that you have to wait $t_a$ time periods until $X \geq 1$"?

(Where $X \geq 1$ is the same things as saying "$X$ has occurred", for example "the component is dead.")

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Strongly related: Exponential distribution from Poisson –  Jacob Akkerboom Mar 11 at 11:11
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Not quite. You need to talk about a Poisson process, not just a single Poisson random variable. If $X_t$ is a Poisson process, then the first time at which $X_t = 1$ is an exponential random variable.

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Does $T$ measure how long I need to wait, in order for the event to happen? (Where "the event to happen" means $X \geq 1$.) –  Lindberg Dec 4 '12 at 21:34
    
Yes, that's what I said. It's how long you have to wait from $t=0$ until $X_t = 1$ (which is almost surely the same as $X_t \ge 1$ because the jumps happen one-at-a-time). –  Robert Israel Dec 4 '12 at 22:58
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