# Poisson process question

1. Mar 2, 2010

### rsq_a

I'm a bit frustrated with this one...

Let $(X_t)_{t\geq 0}$ be a Poisson Process with rate $\lambda$

Each time an 'arrival' happens, a counter detects the arrival with probability $p$ and misses it with probability $1-p$. What is the distribution of time, $T$ until the first particle is detected?

I thought I could do it like this:

$$P(T = t) = \sum_{n=1}^\infty P(T = t | X_t = n) P(X_t = n) = \sum_{n=1}^\infty (1-p)^{n-1}p \frac{e^{-\lambda t}(\lambda t)^n}{n!}$$

Unfortunately, this does give me the right answer (T should be Exponentially distributed with parameter $\lambda p$). I've stared at my process, but I can't seem to find anything wrong with it!

Can someone point out the error in my approach?