A store promises to give a small gift to every 13th customer to arrive. If the arrivals of customers form a Poisson process with rate $\lambda$,

a) find the probability density function of the times between the lucky arrivals;

b) find $P(M_t = k)$ for the number of gifts $M_t$ given in the interval $[0, t]$.


A typical Poisson process denoted by $N_t$ refers to the number of arrivals (discrete) at time $t$, with rate $\lambda$, where

$$P(N_t = k) = \frac{e^{\lambda t} (\lambda t)^k}{k!}$$

  • 1
    $\begingroup$ Have you learned about the Erlangian distributions? $\endgroup$
    – Brian Tung
    Commented Jul 22, 2019 at 3:48

1 Answer 1


Would you use the Erlang-n function with n = 13?

In that case the density would be $$\frac{\lambda (\lambda t)^{12} e^{-\lambda t}}{12!}$$


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