# Conditional Distribution of arrival times

• Dec 10th 2011, 08:22 AM
puggles
Conditional Distribution of arrival times
Just got my final back, and this was the only problem I couldn't figure out. Could someone show me how to do this? Thanks!

A Poisson process N(t) has rate $\lambda$= 1. Let $X_2$ be the time between the first and second arrivals of the process. Suppose there are exactly 2 arrivals in the interval $0\leq t\leq 1$, so N(1) = 2. Find the conditional pdf of $X_2$ given that N(1) = 2. Use your result to calculate the mean and standard deviation of $X_2$ given that N(1) = 2.

We were given the hint to use the theorem of conditional arrival times, namely that given N(t) = n, the n arrival times $S_1,...,S_n$ have the same distribution as the order statistics corresponding to n independent random variables uniformly distributed on (0,t).

The only headway I was able to make was that $X_2 = S_2 - S_1$, so the conditional pdf of $X_2$ would be the pdf of $S_2$ minus the pdf of $S_1$. But when I used that to find the expected value, I got a ln(t), which is undefined at 0 so I figured I was wrong.
• Dec 14th 2011, 06:46 AM
chisigma
Re: Conditional Distribution of arrival times
The answer to that question is strongly related to the following theorem in...

http://www.nas.its.tudelft.nl/people...UP_Poisson.pdf

Theorem 7.3.3 Given that exactly one event of a Poisson process $\{X(t);\ t \ge
0\}$
has occurred during the interval [0, t], the time of occurrence of this event is uniformly distributed over [0, t].

Naw if $T_{1}$ and $T_{2}$ are two two times of occurrence in [0,1], then the random variable $\Delta T= |T_{2}-T_{1}|$ has p.d.f. given by...

$p(\tau)=\begin{cases}2\ (1-\tau)} &\text{if}\ 0<\tau<1\\ 0 &\text{elsewhere}\end{cases}$ (1)

http://www.sv-luka.org/ikone/ikone180a.jpg

Marry Christmas from Serbia

$\chi$ $\sigma$