# Conditional Poisson process

I am having difficulty with the following problem: A store promises to give a small gift to every thirteenth customer to arrive. If the arrivals of customers form a Poisson process with rate $\lambda$, then:

1. Find the probability density function of the times between the lucky arrivals;
2. Find $P[M(t) = k ]$ for the number of gifts $M(t)$ given during the interval $[0, t]$
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What do you know? What did you try? Where are you stuck? –  Did May 24 '12 at 18:49
my problem is the use of the condition given by the number 13 –  hugo ramires May 24 '12 at 18:55
??? Please explain. –  Did May 24 '12 at 19:09

1. The customer interarrival time is exponential $\mathcal{E}(\lambda)$. 13 customers need to arrive, the time between lucky arrivals is thus the sum of 13 independent identically distributed exponential random variables. This sum follows a well-known distribution.
2. $\mathbb{P}(M(t)=k) = \mathbb{P}(13 k \leqslant N(t) < 13(k+1))$, now use what you know about the Poisson process $N(t)$.