Poisson Distribution
The Poisson distribution models event counts over a fixed exposure when events occur at a stable average rate. It is the first actuarial model for claim frequency, arrivals, and rare-event approximations.
What Poisson answers
Poisson answers a count question: how many events occur in a fixed exposure period? The exposure might be one policy-year, one month of claim reporting, one mile of road, or one hour of calls.
The model is most natural when events are individually rare, many opportunities exist, and the average rate is stable over the exposure being modeled. In actuarial language, it is a frequency model before severity is brought in.
Core formulas
If the expected number of events is lambda, the probability of exactly k events is given by the mass function below. For Poisson, the mean and variance are equal, which makes overdispersion an important model check later.
Claim-count worked example
If a portfolio averages 2 claims per month, the probability of exactly 3 claims next month is about 0.180. The same setup also gives the chance of no claims: e to the negative 2, about 0.135.
Those two probabilities tell different business stories. Exactly 3 claims is a point probability. No claims is the start of a tail calculation because it helps answer at least one claim, at least two claims, and stop-loss questions.
Run Poisson claim-count calculations
Use the same function for exact counts, zero claims, and exposure-scaled means. Adjust the monthly rate or exposure window to test nearby scenarios.
Exactly 3 claims next month: 0.1804 At least 1 claim next month: 0.8647 Mean for 4 months at 1.5 per month: 6.0
Exposure scaling
Rates scale with exposure. If a portfolio averages 2 claims per month, it averages 6 claims over 3 months when the rate is stable. The distribution changes because lambda changes with the exposure length.
Do not mix rate and exposure. A rate of 2 per month is not the same as a Poisson mean of 2 unless the exposure window is one month.
Poisson process connection
A Poisson count model and an exponential waiting-time model are two views of the same arrival story. Poisson counts events in a time interval. Exponential measures the waiting time to the next event.
This connection is high-yield because Exam P can give arrival language and ask for either a count probability or a waiting-time probability.
Exam traps
Most Poisson mistakes come from using the wrong exposure, treating the rate as a probability, or forgetting that at least one event is easier to calculate by complement.
- If the problem says per year but asks for a quarter, rescale lambda before using the mass function.
- For at least one event, use 1 minus the probability of zero events.
- For independent Poisson counts, sums are Poisson with lambdas added.
- If observed variance is much larger than the mean, Poisson may be too narrow for modeling even when it remains exam-relevant.
Where Poisson connects next
Exponential waiting times explain the arrival-process side. Poisson approximation to binomial explains the rare-event limit. Compound Poisson adds severity to frequency, turning claim counts into aggregate losses.
Original exam practice
4 questions built from syllabus outcomes and released-exam patterns. The prompts and answers are original, so they train the skill without copying official exam text.
Poisson Claim Frequency Drill
Original Poisson checks for count probabilities, exposure scaling, complements, and independent sums.
- Question 1/Calculation
Exactly three claims
A portfolio averages 2 claims per month. Assuming a Poisson count model, find the probability of exactly 3 claims next month.
Corepoisson-distributionclaim-frequencyexact-countexam-pPoisson DistributionExponential DistributionSolution and grading points
Use lambda equal to 2 for one month. The probability is e to the negative 2 times 2 cubed divided by 3 factorial, which is about 0.180.
- Use the one-month expected count as the Poisson mean.
- Evaluate the Poisson mass function at 3.
- Keep the result as a probability, not an expected claim count.
- Uses lambda equal to the one-month expected count.
- Uses the Poisson mass function for exactly 3 claims.
- Reports a probability near 0.180.
- Question 2/Calculation
Quarterly exposure
A claim process averages 1.5 claims per month. Under a stable Poisson rate, what mean should be used for a 4-month count?
Solution and grading points
Scale the rate by exposure. Four months at 1.5 claims per month gives lambda equal to 6 for the 4-month count.
- Recognizes that the rate must be scaled by exposure.
- Multiplies 1.5 by 4.
- Uses 6 as the Poisson mean for the longer period.
- Question 3/Calculation
At least one claim
A small block has a Poisson mean of 0.4 claims over a week. Find the probability of at least one claim during the week.
Solution and grading points
Use the complement. The probability of no claims is e to the negative 0.4, so the probability of at least one claim is 1 minus that value, about 0.330.
- Uses the complement of zero claims.
- Computes the zero-claim probability with lambda equal to 0.4.
- Reports a probability near 0.330.
- Question 4/Written Answer
Independent count sum
Two independent portfolios have Poisson means 3 and 5 for the same month. What is the distribution of the combined monthly claim count?
Solution and grading points
The combined count is Poisson with mean 8 because independent Poisson counts add by adding their means.
- Uses independence.
- Adds the two Poisson means.
- Names the combined distribution as Poisson with mean 8.