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Poisson Distribution worksheets

Scale a constant event rate to an exposure, then calculate exact, cumulative, tail, mean, and variance results for Poisson counts.

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Assume events occur independently at a constant average rate. Convert rate and exposure to lambda in matching units. Use a calculator for exponentials and Poisson sums. Round final probabilities to the configured 3 or 4 decimal places; use scientific notation when a rounded display would be 0.

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Keep the rate and units visible

Poisson sheets distinguish a rate per unit from lambda for the full exposure. Exponential sheets label lambda as a positive rate and connect it to the reciprocal mean waiting time.

Learning objective

What this activity builds

Translate rate per unit and exposure into the Poisson mean lambda, then use the correct count probability or moment formula.

Worked example

Calls arrive at 2 per hour. Over 3 hours, what is the mean count?

lambda = (2 calls/hour)(3 hours) = 6 calls, so E(X)=6.

A Poisson rate scales linearly with the amount of time, distance, or area observed.

Teaching notes

Make the practice count

  • Write rate units beside every conversion.
  • Distinguish the rate per unit from lambda for the full exposure.
  • For at least k, identify whether a complement shortens the work.

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Math SheetNameScoreTeacherDatePoisson DistributionAssume events occur independently at a constant average rate. Convert rate and exposure to lambda in matching units. Use a calculator for exponentials and Poisson sums. Round finalprobabilities to the 3 decimal places; if that would display 0, use scientific notation.1.Arrivals occur independently at a constant average rate of 0.4 per hour. Observe 0.75 hours. Find the probability of at least 6 arrivals.Assumptions: The arrivals occur independently with the same constant average rate throughout the stated hour exposure; disjoint exposure segments have independent counts.Reference: lambda = (rate per unit)(exposure), and P(X=k) = exp(-lambda) lambda^k / k!. Calculator required for exponential values and Poisson sums.2.Arrivals occur independently at a constant average rate of 0.4 per hour. Observe 1.5 hours. Find the probability of exactly 7 arrivals.Assumptions: The arrivals occur independently with the same constant average rate throughout the stated hour exposure; disjoint exposure segments have independent counts.Reference: lambda = (rate per unit)(exposure), and P(X=k) = exp(-lambda) lambda^k / k!. Calculator required for exponential values and Poisson sums.Keep rate and exposure units consistent. Round only the final result.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDatePoisson DistributionAssume events occur independently at a constant average rate. Convert rate and exposure to lambda in matching units. Use a calculator for exponentials and Poisson sums. Round finalprobabilities to the 3 decimal places; if that would display 0, use scientific notation.3.Arrivals occur independently at a constant average rate of 0.25 per hour. Observe 2 hours. Find the mean and population variance of the number ofarrivals. State the count support and distinguish this Poisson count from an exponential waiting time.Assumptions: The arrivals occur independently with the same constant average rate throughout the stated hour exposure; disjoint exposure segments have independent counts.Reference: lambda = (rate per unit)(exposure), and P(X=k) = exp(-lambda) lambda^k / k!. Calculator required for exponential values and Poisson sums.4.Surface flaws occur independently at a constant average rate of 1.25 per kilometer. Observe 0.5 kilometers. Find the probability of at most 7 surfaceflaws.Assumptions: The surface flaws occur independently with the same constant average rate throughout the stated kilometer exposure; disjoint exposure segments have independentcounts.Reference: lambda = (rate per unit)(exposure), and P(X=k) = exp(-lambda) lambda^k / k!. Calculator required for exponential values and Poisson sums.Keep rate and exposure units consistent. Round only the final result.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDatePoisson DistributionAssume events occur independently at a constant average rate. Convert rate and exposure to lambda in matching units. Use a calculator for exponentials and Poisson sums. Round finalprobabilities to the 3 decimal places; if that would display 0, use scientific notation.5.Surface flaws occur independently at a constant average rate of 1.5 per kilometer. Observe 2 kilometers. Find the probability of exactly 6 surfaceflaws.Assumptions: The surface flaws occur independently with the same constant average rate throughout the stated kilometer exposure; disjoint exposure segments have independentcounts.Reference: lambda = (rate per unit)(exposure), and P(X=k) = exp(-lambda) lambda^k / k!. Calculator required for exponential values and Poisson sums.6.Material defects occur independently at a constant average rate of 3 per square meter. Observe 4 square meters. Find the mean and populationvariance of the number of material defects. State the count support and distinguish this Poisson count from an exponential waiting time.Assumptions: The material defects occur independently with the same constant average rate throughout the stated square meter exposure; disjoint exposure segments haveindependent counts.Reference: lambda = (rate per unit)(exposure), and P(X=k) = exp(-lambda) lambda^k / k!. Calculator required for exponential values and Poisson sums.Keep rate and exposure units consistent. Round only the final result.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDatePoisson DistributionAssume events occur independently at a constant average rate. Convert rate and exposure to lambda in matching units. Use a calculator for exponentials and Poisson sums. Round finalprobabilities to the 3 decimal places; if that would display 0, use scientific notation.7.Arrivals occur independently at a constant average rate of 1 per hour. Observe 2 hours. Find the probability of at least 0 arrivals.Assumptions: The arrivals occur independently with the same constant average rate throughout the stated hour exposure; disjoint exposure segments have independent counts.Reference: lambda = (rate per unit)(exposure), and P(X=k) = exp(-lambda) lambda^k / k!. Calculator required for exponential values and Poisson sums.8.Arrivals occur independently at a constant average rate of 2.5 per hour. Observe 1.5 hours. Find the probability of at most 1 arrivals.Assumptions: The arrivals occur independently with the same constant average rate throughout the stated hour exposure; disjoint exposure segments have independent counts.Reference: lambda = (rate per unit)(exposure), and P(X=k) = exp(-lambda) lambda^k / k!. Calculator required for exponential values and Poisson sums.Keep rate and exposure units consistent. Round only the final result.math-sheet.comPage 4 of 4

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