TheIBAxolotl
When to use Poisson
The Poisson distribution models how many times an event happens in a fixed interval of time, space or volume: calls to a help desk per hour, typing errors per page, bacteria per millilitre. The model fits when:
- events occur randomly and independently of each other,
- they occur at a constant average rate,
- two events cannot happen at exactly the same instant.
If \(X\) is the number of events and the mean in the interval is \(m\), write \(X \sim Po(m)\).
The probabilities
\[P(X = x) = \frac{m^x e^{-m}}{x!}, \quad x = 0, 1, 2, \dots\]
There is no upper limit on \(x\), unlike the binomial distribution.
Mean equals variance
For a Poisson variable,
\[E(X) = \operatorname{Var}(X) = m\]
This gives a quick test for real data: if the sample mean and variance are close, Poisson may be a sensible model; if the variance is much larger, events are probably clustering and the model is doubtful.
Scaling the rate
The mean is proportional to the interval length. If calls arrive at 4 per 10 minutes, then over 5 minutes \(m = 2\), and over an hour \(m = 24\). Always rescale before calculating.
Sums of Poisson variables
If \(X \sim Po(a)\) and \(Y \sim Po(b)\) are independent, then \(X + Y \sim Po(a + b)\). Two independent help desks receiving 3 and 5 calls an hour together receive \(Po(8)\) calls an hour.
Worked example
Calls arrive at a help desk at an average of 4 every 10 minutes.
- Exactly 2 calls in 5 minutes: \(m = 2\), \(P(X = 2) = \dfrac{2^2 e^{-2}}{2!} = 2e^{-2} \approx 0.271\).
- At least 3 calls in 10 minutes: \(m = 4\), \(P(X \ge 3) = 1 - P(X \le 2) = 1 - e^{-4}(1 + 4 + 8) \approx 0.762\).
- Over 8 separate 10-minute intervals, the number with at least 3 calls follows \(B(8, 0.762)\); questions often chain Poisson into binomial like this.
GDC tips
- Poisson pdf gives \(P(X = x)\); Poisson cdf gives \(P(X \le x)\).
- Convert every inequality to "\(\le\)" form: \(P(X < 5) = P(X \le 4)\), \(P(X > 5) = 1 - P(X \le 5)\), \(P(X \ge 5) = 1 - P(X \le 4)\).
- To find an unknown mean, for example \(P(X = 0) = 0.2\), solve \(e^{-m} = 0.2\) to get \(m = \ln 5 \approx 1.61\).
Common mistakes
- Forgetting to rescale \(m\) to the new interval.
- Using \(1 - P(X \le 3)\) for "at least 3" (it should be \(1 - P(X \le 2)\)).
- Applying Poisson when events are clearly not independent (customers arriving in groups).
- Adding standard deviations instead of means when combining variables.
Exam tips
- State the distribution with its parameter, such as \(X \sim Po(2)\), before any calculation.
- When asked to justify a Poisson model, quote a condition in context ("calls arrive independently at a constant average rate").
- Give probabilities to 3 significant figures.