Poisson Distribution Table
Explore exact and cumulative Poisson probabilities in a searchable table. Select the average event rate (λ) and event count (k), then read the highlighted probability with an explanation of its meaning.
Poisson Probability Distribution Table
A Poisson distribution table shows probabilities for different event counts and average rates. Each row represents an event count k, while each column represents an expected count λ for the same interval.
Switch between PMF for exact probabilities and CDF for cumulative probabilities.
Supported inputs: λ from 0 to 10,000 and integer k from 0 to 100,000. Custom λ appears as its own column. Change k to locate any supported count; large counts use a 31-row window.
For λ = 3, the chance of exactly two events is approximately 22.40%.
≈ marks rounded values. Very small probabilities use scientific notation; numeric limits do not mean mathematical impossibility or certainty. CDF values always include the full lower tail, not just visible rows.
Event counts 0–15 shown; the Poisson support continues to infinity.
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How to Read a Poisson Distribution Table
First, identify the average number of events expected in the interval you are studying. This is λ. Next, find the event count k. Select the appropriate probability view, then read the value at the intersection of the λ column and k row.
If the average rate and the question use different interval lengths, adjust λ before reading the table.
Poisson PMF and Cumulative Distribution
PMF gives P(X = k), the probability of exactly k events.
CDF gives P(X ≤ k), the probability of no more than k events.
For more than k events, use P(X > k) = 1 − P(X ≤ k). For at least k events, use P(X ≥ k) = 1 − P(X ≤ k − 1).
The lookup evaluates upper tails directly through the existing incomplete-gamma engine rather than subtracting rounded CDF cells.
Poisson Distribution Formula
P(X = k) = e^(−λ) λ^k / k!
The symbol λ represents the expected count over the selected interval, and k! means k factorial. For the Poisson distribution, both the mean and variance equal λ.
The model assumes events follow a suitable independent, constant-rate process.
P(X ≤ k) = Σᵢ₌₀ᵏ e^(−λ) λⁱ / i!; standard deviation = √λ.
Worked Examples Using the Poisson Table
These examples use sample values, not your current inputs.
Example 1 — Exactly two events
A system receives an average of three events per interval. With λ = 3:
P(X = 2) = e^(−3) × 3² / 2! ≈ 0.224042.
The probability of exactly two events is approximately 22.40%.
Example 2 — At most two events
Using the same average rate:
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2) ≈ 0.423190.
The probability of two or fewer events is approximately 42.32%.
Example 3 — Adjusting the interval
If a call center receives six calls per hour on average, a 30-minute interval has λ = 6 × 0.5 = 3, assuming a constant rate.
Use λ = 3 rather than λ = 6 when looking up probabilities for that half-hour interval.
When Is a Poisson Distribution Appropriate?
A Poisson model is useful for counts of events over a fixed period, area, distance, or other interval when the underlying process has a constant expected rate and suitable independence assumptions.
It may be inappropriate for strongly clustered events, changing rates, or data with substantially greater variance than the mean.
For a fixed number of independent trials, compare the Binomial Distribution Table.
Frequently Asked Questions
What is a Poisson distribution table?
A Poisson distribution table lists probabilities for counts of events at different average rates. It can display exact probabilities P(X = k) or cumulative probabilities P(X ≤ k).
How do I read a Poisson distribution table?
Find the column for λ, the average number of events in your interval, and the row for k, the event count. The intersecting cell gives the probability according to the selected PMF or CDF view.
What does lambda mean in a Poisson distribution?
Lambda (λ) is the expected number of events over the interval being modeled. For example, an average of four events per hour gives λ = 4 for one hour and λ = 2 for half an hour, assuming a constant rate.
What is the difference between Poisson PMF and CDF?
PMF gives the probability of exactly k events. CDF gives the probability of k or fewer events, including k.
How do I find the probability of more than k events?
Use the complement of the cumulative probability: P(X > k) = 1 − P(X ≤ k). The phrase “more than k” excludes k.
Why can a Poisson table show very small probabilities?
Events far from the expected count may have very small probabilities. A value displayed as 0.0000 may be a rounded positive value rather than exactly zero. This table uses scientific notation for small positive probabilities.
When can Poisson approximate a binomial distribution?
A Poisson approximation can be useful when a binomial model has many trials and a small success probability. Set λ = np and assess whether the approximation is sufficiently accurate for the values and event probabilities involved.
What happens when lambda equals zero?
If λ = 0, the model assigns probability 1 to zero events and probability 0 to every positive event count. This is a degenerate Poisson distribution.
Sources and methodology
NIST — Poisson Distribution defines the PMF, CDF and moments. This lookup uses our existing log-gamma PMF and regularized incomplete-gamma tails, avoiding factorial overflow. At λ = 0 the distribution is a point mass at zero. Floating-point underflow is labeled, not presented as exact zero.
Poisson Distribution Calculator supports event ranges and full calculations. Browse all tables or read our methodology.