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I've a problem about poisson distribution.

Let $X$ denote The number of vehicles arrive at a garage every hour. $X$ is a random variable evaluated by Poisson distribution with $λ$ as the distribution parameter. At the garage,There are $M≥1$ available repair stations and the service duration of each station is exactly an hour. If a vehicle arrives and all the stations are occupied,it immediately leaves the garage without being repaired.In the beginning of the day,all the stations are available.

calculate the probability functions of the following random variables :

  1. $Y$-the number of vehicles that got service at the first hour of the day.

  2. $Z$-the number of vehicles that left the garage without being fixed at the first hour of the day.

The two functions should be represented as a sum.

If anything about this question is vague,let me know .

Thanks a lot for your help !

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1 Answer 1

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Sketches:

  1. $Y=\min (X,M)$, so $\Pr(Y=k)=\Pr(X=k)$ for $k<M$ and $\Pr(Y=M)=\Pr(X\geq M)$.
  2. $Z=X-Y$, so $\Pr(Z=0)=\Pr(X=0)+...+\Pr(X=M)$ and $\Pr(Z=k)=\Pr(X=M+k)$ for $k\geq 1$.
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