BYTETOOLS

Poisson Distribution Calculator

Calculate Poisson probabilities for a given rate: P(X = k), cumulative tails, range probabilities, mean and variance, with a bar chart and full probability table.

0.180447
P(X = 3)
0.676676
P(X < 3)
0.857123
P(X ≤ 3)
0.142877
P(X > 3)
0.323324
P(X ≥ 3)

Distribution summary

2
Mean (λ)
2
Variance (λ)
1.414214
Std deviation (√λ)
2
Mode ⌊λ⌋

P(1 ≤ X ≤ 4) = 0.8120117 (81.2012%)

P(X = k) = λᵏ · e^(−λ) / k! is evaluated in log space with a log-gamma factorial, and the cumulative probability uses the identity P(X ≤ k) = Q(k+1, λ), so very large λ and k stay exact instead of overflowing. Worked check: λ = 2, k = 3 gives 0.180447.

Probability mass (k = 3 highlighted)

0: 0.13533528321: 0.27067056652: 0.27067056653: 0.18044704434: 0.090223522165: 0.036089408866: 0.012029802957: 0.0034370865588: 0.00085927163969: 0.000190949253210: 0.00003818985065012345678910

Probability table

kP(X = k)P(X ≤ k)
00.135335280.13533528
10.270670570.40600585
20.270670570.67667642
30.180447040.85712346
40.090223520.94734698
50.036089410.98343639
60.01202980.99546619
70.003437090.99890328
80.000859270.99976255
90.000190950.9999535
100.000038190.99999169

What is the Poisson Distribution Calculator?

The ByteTools Poisson Distribution Calculator works out how likely a given number of events is when they happen independently at a known average rate.

  • P(X = k) plus P(X < k), P(X ≤ k), P(X > k) and P(X ≥ k)
  • Range probability P(a ≤ X ≤ b) between any two counts
  • Mean, variance, standard deviation and mode
  • Log-space evaluation so large λ and k never overflow
  • Bar chart with your k highlighted and a scrollable probability table
  • Entirely browser-based — no data leaves your device

How to use the Poisson Distribution Calculator

  1. 1

    Enter the average rate λ — the mean number of events per interval.

  2. 2

    Enter the number of occurrences k you want the probability for.

  3. 3

    Optionally set a lower and upper bound to get the probability of a range.

  4. 4

    Read P(X = k) plus the four cumulative probabilities across the top.

  5. 5

    Check the mean, variance and mode, then use the chart and table for the full distribution.

About the Poisson Distribution Calculator

The ByteTools Poisson Distribution Calculator works out how likely a given number of events is when they happen independently at a known average rate. Enter the rate λ and a count k, and you get P(X = k) together with all four cumulative probabilities and the probability of landing inside a range.

The Poisson distribution models call-centre arrivals, website visits per minute, typos per page, machine failures per month and radioactive decays — anything counted over a fixed interval. Because the mean and variance are both λ, the summary panel makes that distinctive property obvious, and the bar chart plus probability table show the whole shape rather than a single number.

Probabilities are evaluated in log space using a log-gamma factorial, and the cumulative values use the incomplete gamma identity P(X ≤ k) = Q(k+1, λ), so very large rates and counts stay accurate instead of overflowing to infinity. Everything runs inside your browser, nothing is uploaded, and the page works offline.

Frequently asked questions

When should I use the Poisson distribution?

Use it for counting how many times something happens in a fixed interval of time, area or volume, when events occur independently at a roughly constant average rate. Classic examples are emails per hour, defects per metre of cable and customers arriving per minute.

What is the formula for the Poisson distribution?

P(X = k) equals lambda to the power k, times e to the minus lambda, divided by k factorial. For example with λ = 2 and k = 3 the probability is 2³ × e⁻² ÷ 6, which comes to 0.1804 or about 18%.

Why are the mean and the variance both equal to lambda?

It falls out of the maths of the distribution, and it is a useful diagnostic. If your real count data has a variance clearly larger than its mean, the data is overdispersed and a negative binomial model usually fits better than a Poisson one.

What is the difference between the Poisson and the binomial distribution?

The binomial counts successes in a fixed number of trials with a known success probability. The Poisson counts events with no fixed number of trials, just an average rate. Poisson is in fact the limit of the binomial as the number of trials grows large and the per-trial probability shrinks.

Can k be larger than lambda?

Yes. λ is only the long-run average, so any non-negative whole number is possible. The probability simply gets smaller the further k moves away from λ, which is why the bar chart tails off on both sides of the peak.

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