Poisson Distribution Calculator (poissonpdf & poissoncdf)

This free Poisson distribution calculator finds the probability of exactly, at most, fewer than, at least, more than or between a number of events when they happen at an average rate λ (lambda). It shows the step-by-step working with your numbers, a bar chart with the outcomes you asked for highlighted, the full poissonpdf / poissoncdf table, the mean, variance and standard deviation, and the exact TI-84 keystrokes. It can also turn a rate and a time into λ, approximate a binomial, find the inverse (percentile or critical value) and test whether your own count data follow a Poisson distribution.

Contents

How to Use the Poisson Distribution Calculator

1. Enter λ, the average number of events in one interval. If you know a rate instead (12 calls per hour) and a different interval (15 minutes), choose Rate × interval and the calculator multiplies them for you. Choose From binomial n × p to use the Poisson as an approximation to a binomial.
2. Pick the probability you need: P(X = x), P(X ≤ x), P(X < x), P(X ≥ x), P(X > x) or P(a ≤ X ≤ b), and type x (and b).
3. Read the answer in the purple box. The table under it lists all five probabilities for the same x with the matching TI-84 command, so you can answer a whole homework question at once.
4. Open the extras: step-by-step working, TI-84 keystrokes, a normal-approximation check and the full probability table.
5. Other tabs: Find x (inverse) gives the smallest x that reaches a cumulative probability (or a right-tail critical value); Fit data estimates λ from your counts and runs a chi-square goodness-of-fit test.

Results update as you type. Copy share link saves your inputs in the URL. For a fixed number of trials use the binomial distribution calculator instead.

What Is the Poisson Distribution?

The Poisson distribution models the number of times an event happens in a fixed interval of time, distance, area or volume: calls to a help desk per hour, typos per page, cars through a toll per minute, goals in a football match, radioactive decays per second, defects per metre of cable. X can be 0, 1, 2, 3 … with no upper limit.

A count is Poisson when:

• events happen independently of each other;
• the average rate is constant over the interval (no rush hour hidden inside your hour);
• two events cannot happen at exactly the same instant, and the chance of an event in a tiny sub-interval is proportional to its length.

The single parameter λ is both the mean and the variance. If your counts vary much more than their mean (over-dispersion) a Poisson model will understate the chance of extreme counts; the Fit data tab checks this for you.

Poisson Distribution Formula

P(X = x) = e^−λ · λ^x / x! for x = 0, 1, 2, …

P(X ≤ x) = Σ_k=0..x e^−λ λ^k / k!

Mean μ = λ, variance σ² = λ, standard deviation σ = √λ

e ≈ 2.71828 and x! is x factorial (0! = 1). The first formula is the probability mass function (PMF), which the TI-84 calls poissonpdf; the second is the cumulative distribution function (CDF), poissoncdf. The most likely count (the mode) is ⌊λ⌋; when λ is a whole number, λ − 1 and λ are equally likely.

Worked Example: λ = 4 Calls per Hour

The calculator opens with this problem: a help desk receives on average 4 calls per hour. What is the probability of exactly 3 calls in the next hour?

1. λ = 4, x = 3. e⁻⁴ = 0.0183156.
2. P(X = 3) = e⁻⁴ × 4³ / 3! = 0.0183156 × 64 / 6 = 0.1954 (about 19.5%).
3. At most 3: P(X ≤ 3) = P(0) + P(1) + P(2) + P(3) = 0.0183 + 0.0733 + 0.1465 + 0.1954 = 0.4335.
4. Fewer than 3: P(X < 3) = P(X ≤ 2) = 0.2381.
5. At least 3: P(X ≥ 3) = 1 − P(X ≤ 2) = 1 − 0.2381 = 0.7619.
6. More than 3: P(X > 3) = 1 − P(X ≤ 3) = 1 − 0.4335 = 0.5665.
7. No calls at all: P(X = 0) = e⁻⁴ = 0.0183, under 2%.
8. Mean = variance = 4 calls, standard deviation = √4 = 2 calls. 3 and 4 calls are equally likely (0.1954 each).

x	0	1	2	3	4	5	6
P(X = x)	0.0183	0.0733	0.1465	0.1954	0.1954	0.1563	0.1042

Rate × Interval: Getting λ Right

λ must be the average for the same interval the question asks about. If a call centre gets 12 calls per hour and you want the chance of at least 5 calls in a 15-minute window, then

λ = rate × interval = 12 × 0.25 = 3

and P(X ≥ 5) = 1 − P(X ≤ 4) = 1 − 0.8153 = 0.1847. Using λ = 12 here is the single most common Poisson mistake. Choose Rate × interval in the calculator (and press Example) to see this case worked out.

At Least, More Than and Between

Because a Poisson count has no maximum, “at least” and “more than” are always found with the complement:

• P(X ≥ x) = 1 − P(X ≤ x − 1) — on a TI-84, 1-poissoncdf(λ,x-1)
• P(X > x) = 1 − P(X ≤ x) — 1-poissoncdf(λ,x)
• P(X < x) = P(X ≤ x − 1) — poissoncdf(λ,x-1)
• P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1) — poissoncdf(λ,b)-poissoncdf(λ,a-1)

For λ = 4, P(2 ≤ X ≤ 6) = P(X ≤ 6) − P(X ≤ 1) = 0.8893 − 0.0916 = 0.7977. “At least one” is the quickest of all: P(X ≥ 1) = 1 − e^−λ.

Poisson Approximation to the Binomial

When a binomial has many trials and a small success probability, the Poisson with λ = np is an excellent shortcut. The usual rule of thumb is n ≥ 20 and p ≤ 0.05 (or n ≥ 100 and np ≤ 10).

Example: 1,000 items, each defective with probability 0.004. λ = 1000 × 0.004 = 4.

Probability	Exact binomial (n = 1000, p = 0.004)	Poisson (λ = 4)
P(X = 3)	0.19556	0.19537
P(X ≤ 3)	0.43308	0.43347

Choose From binomial n × p and the calculator shows the exact binomial answer next to the Poisson one, says whether the rule of thumb is met, and gives the binomcdf command so you can check it on a TI-84.

Inverse Poisson: Percentiles and Critical Values

Sometimes you know a probability and need the count. Counts are whole numbers, so the inverse is defined as the smallest x that reaches the target:

• Percentile: smallest x with P(X ≤ x) ≥ p. For λ = 4 and p = 0.95, P(X ≤ 7) = 0.9489 is just short and P(X ≤ 8) = 0.9786, so the 95th percentile is 8. A store that averages 4 sales an hour needs stock for 8 to cover 95% of hours.
• Right-tail critical value: smallest x with P(X ≥ x) ≤ α. For H₀: λ = 6 and α = 0.05, P(X ≥ 10) = 0.0839 but P(X ≥ 11) = 0.0426, so you reject H₀ if you see 11 or more events. The actual size of the test is 0.0426, not exactly 0.05, because counts are discrete.

The TI-84 has no invPoisson command; the calculator shows how to build the cumulative table with seq( and poissoncdf( and read it off.

Poisson Goodness of Fit (Does My Data Follow a Poisson?)

To check whether real data follow a Poisson distribution, open the Fit data tab and enter how many intervals had 0, 1, 2, … events (or paste the raw counts). The calculator:

1. estimates λ̂ = sample mean (or uses a λ you give it);
2. computes the expected frequency for each count, E = N × P(X = x), with the last cell as “x or more”;
3. pools neighbouring cells until every expected count is at least 5;
4. runs the chi-square goodness-of-fit test with df = groups − 1 − 1 (one degree of freedom is lost for estimating λ);
5. checks dispersion: for a Poisson the variance should be close to the mean.

The built-in example is Bortkiewicz’s classic data on Prussian cavalry soldiers killed by horse kicks (200 corps-years: 109 with 0 deaths, 65 with 1, 22 with 2, 3 with 3, 1 with 4). λ̂ = 122 / 200 = 0.61, the variance is 0.611 (ratio 1.002), and after pooling to 0, 1 and “2 or more” χ² = 0.0628 with df = 1, p = 0.802: the Poisson model fits very well. For contingency tables and other distributions, use the chi-square calculator.

Normal Approximation to the Poisson

For large λ the Poisson looks like a bell curve with mean λ and standard deviation √λ. The approximation is reasonable from about λ ≥ 10 and good from λ ≥ 20; always add a continuity correction of 0.5. For λ = 25:

P(X ≤ 30) ≈ P(Y < 30.5) with Y ~ N(25, 5) = 0.8643, versus the exact 0.8633.

The calculator prints this check (and the normalcdf command) under every answer. With modern calculators there is little reason to approximate; it is mostly asked for on exams. See the normal distribution calculator for more.

How to Use poissonpdf and poissoncdf on a TI-84

1. Press 2nd VARS to open the DISTR menu.
2. Scroll to C:poissonpdf( for “exactly x” or D:poissoncdf( for “at most x”. (On a TI-83 Plus or an old TI-84 OS without invT the letters are B and C.)
3. Enter λ at the μ prompt (TI calls the mean μ) and the x value, highlight Paste, and press ENTER twice. For the default example, poissonpdf(4,3) = 0.1953668148 and poissoncdf(4,3) = 0.4334701204.
4. At least 3: type 1 − first, then poissoncdf(4,2) → 0.7618966944.
5. Between 2 and 6: poissoncdf(4,6)-poissoncdf(4,1) → 0.7977.
6. Several x at once: put a list in the x slot, for example poissonpdf(4,{0,1,2,3}).
7. Goodness of fit: put observed counts in L1 and expected counts in L2, then STAT ▸ TESTS ▸ D:χ²GOF-Test. Set df yourself (groups − 2 when λ was estimated from the data).

The results panel prints these keystrokes with your own numbers. Practise on our free TI-84 Plus CE calculator online.

Poisson vs Binomial vs Normal

	Poisson	Binomial	Normal
Counts	events in an interval, no upper limit	successes in n fixed trials	continuous measurements
Parameters	λ	n, p	μ, σ
Mean / variance	λ / λ	np / np(1 − p)	μ / σ²
TI-84	poissonpdf, poissoncdf	binompdf, binomcdf	normalpdf, normalcdf, invNorm
Example	calls per hour	heads in 10 flips	heights of adults

Ask: is there a fixed number of trials? If yes, it is binomial; if you are counting events over time or space with no fixed maximum, it is Poisson.

Common Mistakes

• Not scaling λ to the interval. 12 per hour is λ = 3 for 15 minutes and λ = 24 for 2 hours.
• Using x instead of x − 1 for “at least”. P(X ≥ 5) = 1 − poissoncdf(λ, 4), not 1 − poissoncdf(λ, 5).
• Mixing up poissonpdf and poissoncdf. pdf = exactly x; cdf = x or fewer.
• Forgetting P(X = 0). poissoncdf starts at 0, and P(X = 0) = e^−λ is often the biggest term when λ is small.
• Using a Poisson when the counts have a fixed maximum (out of n trials) — that is binomial, unless n is large and p small.
• Ignoring over-dispersion. If the variance is well above the mean, real data have more extreme counts than the Poisson predicts.
• Leaving the χ²GOF-Test df at the default. When λ is estimated from the data, df = groups − 2.

Why This Poisson Calculator Is Different

• All six probability types, plus every related probability for the same x in one table, each with its TI-84 command.
• Rate × interval and binomial n × p inputs, so you never set up λ wrong, with an exact-binomial comparison and the rule-of-thumb check.
• Inverse Poisson (percentile and right-tail critical value), which most Poisson calculators and the TI-84 itself do not have.
• Goodness-of-fit and dispersion test on your own data, with pooled cells, the correct degrees of freedom and the χ²GOF-Test keystrokes.
• Step-by-step working, a highlighted bar chart and the full poissonpdf / poissoncdf table for any λ up to 1,000,000.
• Accurate in the tails: the probabilities use the same saddle-point method as R’s dpois and are checked against SciPy and 40-digit mpmath values on more than 1,400 cases.
• Dark mode, share links and a phone-friendly layout.

Poisson Distribution Calculator FAQ

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