# First passage time for Compound Poisson distribution

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.

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

\begin{align*} P(T>t) &= 1 - P(T
Since $$P(T>t)$$ equal to the CDF of exponential distribution, therefore, $$T \sim Exp(\lambda)$$