StatisticsdrillPoisson distribution
The Poisson distribution gives the probability of k events in a fixed interval when events happen independently at a constant average rate lambda: P(X = k) = e^(-lambda) * lambda^k / k!. Its mean and variance both equal lambda.
Syntax
P(X = k) = e^(-lambda) * lambda^k / k!| Part | What it is |
|---|---|
lambda | the average number of events per interval (must match the interval in the question) |
k | the count you want the probability of: 0, 1, 2, ... |
e | Euler's number, about 2.71828 |
Examples
A shop gets 2 calls per hour on average. P(no calls in an hour)
P(0) = e^(-2) * 2^0 / 0!Result: 0.1353
Excel: =POISSON.DIST(0, 2, FALSE)
Same rate, exactly 3 calls
P(3) = e^(-2) * 8 / 6Result: 0.1804
lambda = 4, at most 1 event
P(0) + P(1) = e^(-4) * (1 + 4)Result: 0.0916
Excel: =POISSON.DIST(1, 4, TRUE)
Rate 2 per hour, interval of 3 hours
use lambda = 6Result: P(0) = e^(-6) = 0.0025
Common mistakes
- Using the hourly rate for a 3-hour question; scale lambda to the interval first.
- Treating k as continuous; Poisson counts are whole numbers, so P(X < 3) means P(0) + P(1) + P(2).
- Applying Poisson when events cluster or the rate changes over the interval; the constant, independent rate assumption fails.
Practice Poisson distribution
These drills check your answer on live data, free, no account needed.
- Poisson: exactly kThe number of returns per afternoon follows a Poisson distribution with a mean of 1.5. What is the probability of exactly 0? Round to 4 decimal places.
- Poisson: at most kA clinic sees an average of 2 urgent walk-ins per evening, following a Poisson distribution. It can handle 1 without calling in extra staff. What is the probability that there are at most 1 walk-ins tonight? Round to 4 decimal places.
Questions
What is the difference between Poisson and binomial?
Binomial counts successes in a fixed number of trials; Poisson counts events in a fixed interval with no upper limit. When n is large and p small, the binomial is close to Poisson with lambda = np.
What are typical Poisson examples?
Calls to a help desk per hour, defects per meter of cable, website errors per day, customers arriving per minute.
Related
- Binomial distributionThe binomial formula P(X = k) = nCk p^k (1 - p)^(n - k), its mean np and standard deviation sqrt(np(1 - p)), exact worked examples and Excel BINOM.DIST.
- Expected valueExpected value is the probability-weighted average of all outcomes. Formula, worked examples with dice, raffles and bets, and the Excel SUMPRODUCT formula.
- Normal distributionHow to find normal distribution probabilities with z-scores, read the standard normal table, find a percentile cutoff, and use NORM.DIST and NORM.INV.
- Probability rulesThe basic probability rules with dice and coin examples: the addition rule for A or B, the multiplication rule for independent events and the complement rule.