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Poisson Distribution §

A random variable has a Poisson distribution with parameter , written as if:

Note that:

The Poisson is often used as a model for counts of rare events like radioactive decay and traffic accidents.

If and , then

The Poisson distribution is perfect when you want to model how many times a rare, independent event happens over a fixed interval (time, space, area, volume…).

Example §

Number of cars arriving at a toll booth per minute: Imagine a highway toll booth. Cars don’t arrive at perfectly regular times—they show up randomly, but on average, maybe:

cars per minute

The Poisson distribution can answer questions like:

• What’s the probability that exactly 10 cars arrive in the next minute?
• What’s the chance that 0 cars arrive (the booth worker gets a 5-second break)?
• How likely is a traffic spike (20+ cars)?

This matters for staffing, queue management, designing toll infrastructure, and predicting congestion.

Say that cars per minute, let be a random variable that indicates that a spike appains when cars say cars arrive in one minute (the fixed interval is time).

So for Poisson().

The first step is to compute the tail probability, a Poisson tail in this example is . We can approximate this either using the poisson distribution or a normal approximation.

This gives . It is straighforward that so about .

This can be even more intuitive using a normal approximation. For a Poisson random variable with parameter we have that and so the standard deviation will be .

With a reasonable large (not in this example) the poisson distribution is well approximated by a normal distribution.

The value 20, in term of z-score:

So for we would obtain a standard deviations of that is rare because an event is in the 1-3 range of the normal.

A value that is 2.3 standard deviations above the mean is rare in any distribution, including Normal, Poisson, Binomial… so this provides a good intuition.

Warning

The actual probability of was computed using the Poisson CDF, not using the normal distribution. The comparison with a normal distribution was only for intuition, not numerical results