Challenging Poisson Probability problem

A garage has three vans which can be hired out for a day at a time. The total number of requests for the hire of a van, for any particular day, has a Poisson distribution with mean 2. If, for any day, fewer than three requests are received for the hire of a van for that day, the vans used are chosen at random. Find the probability that a particular van is not used on a given day.

Answer: 0.406

Solution poisson

Sum and Difference of Poisson Distributions

For 2 independent Poisson distributions, X~Po(ʎ) and Y~Po(µ)

X+Y ~Po(ʎ+µ)

X-Y does NOT follow Poisson distribution.

If X and Y fulfill the conditions for using Normal approximation to Poisson, then X and Y can be approximated individually using Normal approximation.

For example X~ Po(30) and Y~ Po(20)

X~N(30,30) approximately and Y ~N(20,20) approximately

Then X-Y~N(10,50) approximately. Continuity correction applies.

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