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What a Poisson distribution table tells you
A Poisson random variable counts how many times an event occurs in a specified interval or exposure. Its parameter λ (lambda) is the average count for that same interval. The National Institute of Standards and Technology (NIST) describes the Poisson distribution as a model for the number of events occurring within a given time interval.
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For a count x = 0, 1, 2, …, the probability of exactly x events is:
P(X = x) = e−λ λx / x!
The cumulative probability through x is F(x; λ) = P(X ≤ x), or the sum of the exact probabilities from zero through x. NIST lists the mean as λ, the standard deviation as √λ, and the possible counts as nonnegative integers. These describe the model; they do not establish that a real process follows it. NIST Engineering Statistics Handbook: Poisson Distribution.
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How do I read a Poisson distribution table?
Read the table heading and any notes before choosing a cell. Table layouts differ: some list exact probabilities P(X = x), while others list cumulative probabilities P(X ≤ x). For the illustrative table used here, columns are λ values, rows are event counts x, and each cell is an exact probability. Its values are rounded to four decimal places. Statology: Poisson Distribution Table.
- Set the interval. Identify the time period or exposure in the question, such as calls per 15 minutes or arrivals per hour.
- Choose λ for that interval. Find the column whose average count matches the interval, not a different time period.
- Choose x. Find the row for the number of events in an exact-probability table.
- Read the cell type. At the intersection, use the value as P(X = x) only if the table contains exact probabilities. If it is cumulative, the value means P(X ≤ x).
- Translate the wording. For ranges or thresholds, add exact probabilities or use the cumulative value and its complement, as appropriate.
For example, in the cited table, λ = 4 and x = 6 give P(X = 6) = 0.1042, or about 10.42%. This is a rounded illustrative table value, not an observed real-world frequency.
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Convert common wording into the right probability
For an integer threshold a, a cumulative table gives F(a; λ) = P(X ≤ a). Use the relationships below to include or exclude the boundary correctly.
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1Repair Windows errors before they cause bigger problems2Scan for outdated or missing drivers - takes under a minute3Clear out junk files and repair common Windows errors| Question wording | Probability | Using a cumulative table |
|---|---|---|
| Exactly a | P(X = a) | Use an exact cell, or calculate F(a) − F(a − 1). |
| Fewer than a | P(X < a) | F(a − 1). |
| At most a | P(X ≤ a) | F(a). |
| More than a | P(X > a) | 1 − F(a). |
| At least a | P(X ≥ a) | 1 − F(a − 1). |
“Fewer than three” excludes three, so it includes 0, 1, and 2. “At most three” includes three as well. That one-count difference is why the cumulative lookup changes.
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Worked examples with λ = 4
Exactly six events
Find the cell at λ = 4 and x = 6 in the exact-probability table: P(X = 6) = 0.1042, or 10.42% using the table’s rounded value.
Fewer than three events
Add the cells for zero, one, and two events: 0.0183 + 0.0733 + 0.1465 = 0.2381, or 23.81%. This sum uses the table’s four-decimal rounded entries; calculations from unrounded probabilities can differ slightly.
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At least three events
Use the complement of fewer than three: P(X ≥ 3) = 1 − P(X < 3) = 1 − 0.2381 = 0.7619, or 76.19% from the rounded table values. The complement is convenient because it avoids adding every probability from three upward.
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Match λ to the interval
Lambda must describe the exact period or exposure in the question. If a process averages six calls in two hours and the question asks about a 15-minute interval, scale the average to that interval: 15 minutes is one eighth of two hours, so λ = 6 × 1/8 = 0.75 calls per 15 minutes.
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For this λ = 0.75 example, OpenStax calculates the probability of more than one call as the complement of zero or one call: P(X > 1) = 1 − [P(X = 0) + P(X = 1)] = 0.1733. This worked example illustrates interval scaling and using a complement instead of summing an unbounded upper tail. OpenStax, Introductory Business Statistics 2e: Poisson Distribution.
Check whether the Poisson model fits the question
A table gives probabilities conditional on λ; it does not show that the event rate is constant in the real process. OpenStax’s instructional explanation presents assumptions including a stable event probability or rate, independence, and no simultaneous successes in the modeled intervals. Treat these as conditions for using the model, not guarantees about observed events.
- Define one event clearly and specify the time interval or exposure.
- Use an average count that applies to that same interval.
- Consider whether the event rate is reasonably stable over the interval and whether events can be treated as independent.
- Do not treat a table probability as an empirical measurement or as proof that the assumptions hold.
NIST also lists the standard deviation as √λ. For a noninteger λ, the mode is the largest integer below λ; when λ is an integer, both λ and λ − 1 are modes. These are properties of the Poisson model, not checks that establish its suitability for a particular process. NIST Engineering Statistics Handbook.
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When a printed table is not enough
A printed table may not include the λ value or count you need, or its rounding may be too coarse for the calculation. Statistical software can calculate Poisson probabilities; NIST notes that general-purpose statistical software commonly supports at least some such functions. Confirm whether the function returns an exact probability or a cumulative probability, and use the same endpoint rules shown above. NIST Engineering Statistics Handbook.
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