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Let $S$ follows a Compound Poisson distribution $(S \sim CP(\lambda,F_x(x))$, i.e. $$S = \sum_{i=0}^{N}X_i,$$ where $N\sim Po(\lambda)$ and $X_i \stackrel{iid}{\sim} Exp(1)$.

I know that the first passage time $T \sim Exp(\lambda)$, but I am struggle to get the proof done.

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  • $\begingroup$ What is the question? $\endgroup$ – Foobaz John Mar 14 at 21:58
  • $\begingroup$ I am trying to proof that $T \sim Exp(\lambda)$. $\endgroup$ – VincentN Mar 14 at 22:07
  • $\begingroup$ What is the connection between $N$ and $T$? $\endgroup$ – Kavi Rama Murthy Mar 14 at 23:33
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The proof as follow:

\begin{align*} P(T>t) &= 1 - P(T<t)\\ &= 1 - \frac{1\times e^{-\lambda t}}{1!}\\ &= 1 - e^{-\lambda t} \end{align*}

Since $P(T>t)$ equal to the CDF of exponential distribution, therefore, $T \sim Exp(\lambda)$

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