What is the Poisson distribution used for?

The Poisson distribution models count data - the number of events in fixed intervals of time, space, or volume. It's used for customer arrivals, manufacturing defects, website visits, phone calls, accidents, and any scenario where events occur randomly at a constant average rate.

How do you calculate Poisson probability?

The Poisson probability formula is P(X = k) = (λ^k × e^(-λ)) / k!, where λ is the average rate and k is the event count. The calculator handles factorial and exponential computations. For cumulative probabilities, sum individual terms from 0 to k.

What does lambda (λ) mean in Poisson distribution?

Lambda (λ) is the rate parameter representing the average number of events in the interval. It simultaneously defines the distribution's mean, variance, and overall shape. Higher λ means more frequent events and a more spread-out, symmetric distribution.

When should I use Poisson instead of binomial?

Use Poisson when events are rare (small p), opportunities are numerous (large n), and λ = np is moderate. Poisson is computationally simpler for n ≥ 20 and p ≤ 0.05. It's ideal when you can't specify n precisely but know the average rate λ.

Why is mean equal to variance in Poisson?

The Poisson's mean = variance = λ property follows from its mathematical derivation and reflects the randomness of event occurrence. This equidispersion means spread scales with the rate. Real data with variance much different from mean suggests Poisson may not be appropriate.

Poisson Distribution Calculator

How to use

Drag λ under Parameters, from 0.1 to 20. Bars, statistics and every calculator result update at once. Learn more about the rate parameter
Statistics shows Expected Value, Variance, Std Deviation and Mode for the current parameters; Show Formulas prints the rule under each card. Learn more about the statistics panel
PMF, the first tab above the chart, draws one bar per value k with height $P(X = k)$ ; the red line marks $E[X]$ . Learn more about the PMF chart
CDF replots the same values as the running total $P(X \le k)$ , climbing from 0 to 1. Learn more about CDF values
Table lists every k with $P(X = k)$ and $P(X \le k)$ to six decimals. Learn more about the Table tab
In the Probability Calculator, tap P(X = k), P(X < k), P(X ≤ k), P(X > k) or P(X ≥ k) and type k; the box below shows the probability to six decimals. Learn more about exact probabilities
Tap a range tab, P(a ≤ X ≤ b), P(a < X < b), P(a ≤ X < b) or P(a < X ≤ b), and type a and b; the four differ in which endpoints count. Learn more about range probabilities
Key Properties in the right column: tap a row's ▼ to open its explanation, one at a time; Real-World Applications below lists typical uses. Learn more about key properties and applications

Modify Parameters and See Results

Number of events in a fixed interval

Parameters

Rate Parameter (λ): 3.0Or type λ:(0.1 to 100)

Rate Parameter (λ)

3.00

Average number of events occurring in the fixed interval

Mean = Variance (E[X] = Var(X))

3.00

Unique property: mean equals variance

Standard Deviation (σ)

1.73

Measure of spread, equal to √λ

Statistics

Expected Value

3.0000

Variance

3.0000

Std Deviation

1.7321

Mode

Probability Calculator

Enter value k:

Key Properties

Real-World Applications

Number of customer arrivals per hour at a store
Number of phone calls received at a call center per minute
Number of defects per manufactured item
Number of emails received per day
Number of radioactive decays per second
Number of accidents at an intersection per month

Adjusting the Rate Parameter

Reading the Statistics Panel

Reading the PMF Chart

Understanding CDF Values

Using the Table Tab

The Explorer at Its Opening Parameter

Computing Exact Probabilities

Calculating Cumulative Probabilities

Range Probability Calculations

Key Properties and Real-World Applications

What is the Poisson Distribution?

Mean Equals Variance Property

Poisson as Binomial Approximation

Related Distributions and Calculators

Adjusting the Rate Parameter

DemoParameters and statistics

Step 0 of 4

Use the λ (lambda) slider to set the average rate of events per interval. Values range from 0.1 (very rare events) to 20 (frequent events), representing the expected count in your time or space window.

As you change λ, watch the distribution shape transform. Low λ values create highly right-skewed distributions concentrated near zero. Higher λ values produce more symmetric, bell-shaped distributions approaching normality.

The parameter λ directly determines both the mean and variance - they're always equal in a Poisson distribution. At λ = 5, you expect 5 events on average with variance also equal to 5.

Reading the Statistics Panel

The Statistics panel sits in the left column, under Parameters. Its four cards show Expected Value, Variance and Std Deviation to four decimals, and Mode as a whole number of events. Every card recomputes on each slider move, so the panel always describes the bars in the middle column; the red

E[X]

line on the chart sits at the Expected Value number.

Show Formulas prints the rule under each card:

E[X] = \lambda

\text{Var}(X) = \lambda

\sigma = \sqrt{\lambda}

and

\text{Mode} = \lfloor \lambda \rfloor

. The button then reads Hide Formulas. The numbers are the same either way; the formulas only show where they come from.

At the opening values,

\lambda = 3

, the cards read 3.0000, 3.0000, 1.7321 and 3. Expected Value and Variance are equal, 3.0000 each, and stay equal at every

\lambda

. At

\lambda = 12

the cards read 12.0000, 12.0000, 3.4641 and 12.

The Parameters cards above the panel repeat each value with its symbol and a one-line meaning: Rate Parameter (λ), Mean = Variance (E[X] = Var(X)) and Standard Deviation (σ). For where the formulas come from, see Mean Equals Variance Property.

Reading the PMF Chart

DemoPMF, CDF and Table tabs

Step 0 of 4

The PMF visualization displays bars for k = 0, 1, 2, ... showing P(X = k), the probability of exactly k events occurring. The chart automatically extends to show the relevant range where probabilities are meaningful.

The distribution's peak (mode) occurs at

\lfloor λ \rfloor

- the floor of λ. For λ = 7.5, the mode is at k = 7. When λ is an integer, both k = λ and k = λ-1 are modes with equal probability.

Bar heights decrease rapidly beyond λ + 3√λ, which encompasses about 99.7% of the probability mass. The calculator displays enough range to show where probability becomes negligible.

Understanding CDF Values

The CDF displays P(X ≤ k), climbing from 0 toward 1 in discrete steps. Each step represents adding another outcome's probability, with larger steps at more probable values near λ.

Use the CDF to quickly assess ranges. The probability P(X ≤ 10) can be read directly from the curve at k = 10. The steepness around the mean indicates concentration - steep means most events cluster near λ.

For large λ, the CDF's S-curve shape becomes more pronounced, reflecting the distribution's approach to normality through the Central Limit Theorem.

Using the Table Tab

Table is the third tab above the chart, after PMF and CDF. It replaces the chart with three columns, k, P(X = k) and P(X ≤ k), one row per value on the chart's axis, every value to six decimals. The rows follow the sliders: move one and the whole table recomputes.

Use it where the chart is too small to read. At the opening values,

\lambda = 3

, the table runs from

k = 0

k = 19

. Row 0 is 0.049787, which is

e^{-3}

; rows 2 and 3 tie at 0.224042, and by the last rows

P(X \le k)

shows 1.000000 to six decimals. Raise

\lambda

and the table grows: at

\lambda = 8

it runs to

k = 30

.

The P(X ≤ k) column is the CDF as numbers, and each P(X = k) entry is one bar of the PMF chart. Subtract two neighbouring P(X ≤ k) rows to recover a P(X = k) entry. For a value the table does not list, or a range of values, use the Probability Calculator below the chart.

The Explorer at Its Opening Parameter

\lambda = 3

the chart shows two equally tall bars, at

k = 2

and

k = 3

, both at

P = 0.2240

. The red mean line falls at

E[X] = \lambda = 3

.

That double peak is not a rounding artefact.

Poisson, lambda = 3: PMF above, CDF below

Two equally tall bars, k = 2 and k = 3, both 0.2240 - consecutive Poisson probabilities are in ratio lambda/k, which is exactly 1 at k = lambda. Variance equals the mean, both 3.

Consecutive Poisson probabilities satisfy

\frac{P(k)}{P(k-1)} = \frac{\lambda}{k}

, so the ratio is exactly

1

when

k = \lambda

. Whenever

\lambda

is a whole number the distribution has two modes, at

\lambda - 1

and

\lambda

; here

P(2) = \frac{9e^{-3}}{2}

and

P(3) = \frac{27e^{-3}}{6}

are the same number. Nudge

\lambda

off an integer and one bar wins, with the mode at

\lfloor \lambda \rfloor

.

The tool draws

k = 0

19

and stops early once the probabilities fall below

10^{-10}

; those twenty bars already sum to

1

at full double precision.

Variance equals the mean, both

3

, which is the Poisson's defining fingerprint and the first thing to test before using it. Count data whose variance clearly exceeds its mean is overdispersed, and a Poisson will understate its spread.

Computing Exact Probabilities

DemoPoint and cumulative probabilities

Step 0 of 4

Enter event count k in the P(X = k) row of the Probability Calculator to find P(X = k) using the Poisson formula: (λ^k × e^(-λ)) / k!. The calculator handles the factorial computation automatically.

The exponential term e^(-λ) rapidly decreases as λ increases, while λ^k grows with k. Their balance creates the characteristic probability distribution. For λ = 4, k = 4, you get about 0.195 - the most likely single outcome.

Try calculating P(X = 0) for different λ values. This "zero probability" equals e^(-λ), showing how quickly the chance of no events drops as the rate increases.

Calculating Cumulative Probabilities

Use P(X ≤ k) to find the probability of k or fewer events. This requires summing Poisson probabilities from 0 to k, which the calculator performs efficiently using optimized algorithms.

P(X > k) gives the complement: the probability of more than k events. This equals 1 - P(X ≤ k) and answers questions like "What's the chance of exceeding our capacity of k events?"

The ≥ and > distinction matters: P(X ≥ k) includes k itself, while P(X > k) starts counting from k+1. This one-event difference can be significant for discrete distributions.

Range Probability Calculations

DemoRange probabilities

Step 0 of 4

The P(a ≤ X ≤ b) row of the Probability Calculator computes it by taking F(b) - F(a-1), where F is the CDF. This gives the probability of event count falling within the specified window.

Four boundary options handle inclusivity:
• [a, b] - Include both bounds
• (a, b) - Exclude both bounds
• [a, b) - Include a, exclude b
• (a, b] - Exclude a, include b

Example: For λ = 6, find P(4 ≤ X ≤ 8). This gives the probability of observing between 4 and 8 events inclusive, capturing the central probability mass.

Key Properties and Real-World Applications

The right column holds two lists. Key Properties has 6 rows, each ending in ▼: Support: X ∈ {0, 1, 2, 3, ...}, Mean Equals Variance, Rare Events Approximation, Independent Events, Additive Property, Approximates Normal Distribution. Row titles here are those at the opening values. Tap a row to open its explanation; tap another and the first closes, so one explanation is open at a time; tap the open row to close it.

Mean Equals Variance and Approximates Normal Distribution quote the current

\lambda

, so they change with the slider. Open a row, move a slider, and the numbers in the open explanation follow.

Real-World Applications, below, lists 6 situations the Poisson distribution models, starting with Number of customer arrivals per hour at a store and Number of phone calls received at a call center per minute. This list is fixed text and does not change with the sliders.

The rows are short summaries. The full treatment is in What is the Poisson Distribution?.

What is the Poisson Distribution?

The Poisson distribution models the number of events occurring in a fixed interval (time, space, volume) when events happen independently at a constant average rate λ. It's fundamental for counting rare or randomly timed occurrences.

Three key assumptions: (1) events occur independently, (2) events occur at a constant average rate, and (3) two events cannot occur at exactly the same instant. When these hold, event counts follow a Poisson distribution.

Common applications include customer arrivals, phone call volumes, manufacturing defects, website traffic, radioactive decay counts, and accident frequencies. For theoretical foundations, see Poisson distribution theory page.

Mean Equals Variance Property

A unique feature of the Poisson distribution is that mean = variance = λ. This equidispersion property distinguishes it from other count distributions like negative binomial (overdispersed) or binomial with small p (underdispersed).

When analyzing count data, if sample variance significantly exceeds sample mean, Poisson may not be appropriate - consider negative binomial for overdispersion. If variance is much less than mean, other models may fit better.

The standard deviation equals √λ, meaning the typical deviation from the mean grows as the square root of the rate. For λ = 16, standard deviation is 4, so about 68% of observations fall within [12, 20].

Poisson as Binomial Approximation

The Poisson distribution approximates the binomial distribution when n is large and p is small, with λ = np. Rule of thumb: use Poisson approximation when n ≥ 20 and p ≤ 0.05.

For example, defect probability p = 0.001 across n = 5000 units gives λ = 5. Computing binomial probabilities with these parameters is computationally intensive; Poisson provides excellent approximation with simpler calculations.

This connection explains why Poisson models rare events well - it's the limiting case of binomial as we have many opportunities (large n) for rare outcomes (small p). For comparisons, see binomial vs Poisson approximation.