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.

  • $\begingroup$ 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

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, $T \sim Exp(\lambda)$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.