# conditional expectation value of poisson process

I consider a poisson process with rate $$\lambda$$ . Since the Inter arrival time follows exponential distribution, the expectation value of the inter arrival time, say $$E[X]= \frac{1}{\lambda}$$. If I look particularly for a time interval [0-15s]. Then,

1. What would be the expectation value of the inter arrival time e.g., first, second, third arrivals,......? Would it be $$E[X| t<15]$$

2. What is the expectation value of the inter arrival time if I can it is conditioned over [5s-15s] i.e., $$E[X| 5<t<15]$$?

• To be clear: Are you looking for $\mathsf E(X\mid X\in[s;t))$ ? That is the (conditional) expected inter-arrival time when given that it occurs within a specified interval $s\leqslant X<t$. – Graham Kemp Jul 21 '17 at 9:16
• By the way, the time until the $k$-th arrival where $k>1$ is not distributed the same as the first arrival. – Nap D. Lover Jul 21 '17 at 11:48
• Dear @LoveTooNap29 your comment is very right. Would you please elaborate then the right distribution for the $k^{th}$ arrival where $k>1$? Thanks in advance – Hallian1990 Dec 22 '17 at 19:41
• @Hallian1990 for $k=1,2,...$ the time until the $k$-th arrival of a Poisson process with mean parameter $\lambda$ is Erlang distributed with parameters $k$ and $\lambda$. The mean time for the $k$-th arrival is $k/\lambda$. Note, an Erlang RV is just a gamma RV with an integer shape parameter, also note when $k=1$, the Erlang distribution reduces to the exponential distribution. If you are unfamiliar with this distribution, a quick google or wiki search will remedy that. – Nap D. Lover Dec 22 '17 at 20:03

\begin{align*}\mathsf E(X\mid s\leqslant X\lt t) ~&=~ \dfrac{\int_s^t \lambda x \mathrm e^{-\lambda x}\operatorname d x}{\int_s^t \lambda \mathrm e^{-\lambda x}\operatorname d x} \\[2ex] &=~ \dfrac{(\lambda s+1) \mathrm e^{-\lambda s}-(\lambda t+1)\mathrm e^{-\lambda t}}{\lambda (\mathrm e^{-\lambda s}-\mathrm e^{-\lambda t})}\end{align*}
• Dear Kemp, I want to calculate the expected value of the inter arrival time of the FIRST and SECOND arrivals, by considering a poisson process with rate $\lambda$ and for time $t= [0-10s]$ How to express this conditional expectation value. In other words, how much time (expected) will be elapsed until we get 2 arrivals, conditioning time this time is less than $10s$ – Hallian1990 Jul 22 '17 at 0:38