When should I use Poisson vs binomial?

Use Poisson when counting events in a continuous interval (e.g., calls per hour, defects per square meter). Use binomial for a fixed number of discrete trials with two outcomes (success/failure). For large n with small p, Poisson approximates binomial with λ = np.

Why does mean equal variance in Poisson?

It's a mathematical property of the distribution: both equal lambda. This makes the standard deviation √λ. Real data with overdispersion (variance > mean) suggests non-Poisson distribution (negative binomial more appropriate). Underdispersion suggests binomial or other distributions.

What's the difference between rate and probability?

Rate (λ in Poisson) is per unit time/space (events per hour, defects per cm²). Probability (p in binomial) is per trial (heads on coin flip). Rate can exceed 1 (e.g., 10 calls per minute); probability cannot. Different distributions accommodate these different setups.

When does Poisson approximate binomial?

When n is large (>50), p is small (<0.10), and np is reasonable (typically np < 10). Then binomial(n, p) ≈ Poisson(np). This is useful for rare events with many trials (defects in millions of parts, mutations across genome).

How do I test if my data is Poisson?

Check: variance ≈ mean? Histogram matches Poisson shape? Chi-square goodness-of-fit test against Poisson. For events over time, also check independence (events don't cluster, time between events is exponential). For inflated zeros, use zero-inflated Poisson model.

What if events have a time-varying rate?

Use non-homogeneous Poisson process: λ(t) varies with time. For example, ER arrivals are higher during weekends/nights. Split into stationary intervals or model the time-dependent rate function. Various extensions handle clustering, overdispersion, and other complications.

Poisson Distribution Calculator

Q: What is a Poisson distribution?

A Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, when events occur independently at a constant average rate (lambda). Both mean and variance equal lambda. Used for counting rare events: defects, arrivals, calls, occurrences.

Q: How do I test if my data is Poisson?

Check: variance ≈ mean? Histogram matches Poisson shape? Chi-square goodness-of-fit test against Poisson. For events over time, also check independence (events don't cluster, time between events is exponential). For inflated zeros, use zero-inflated Poisson model.

Q: What if events have a time-varying rate?

Use non-homogeneous Poisson process: λ(t) varies with time. For example, ER arrivals are higher during weekends/nights. Split into stationary intervals or model the time-dependent rate function. Various extensions handle clustering, overdispersion, and other complications.

Enter the average rate (lambda) and the number of events (k) to compute exact and cumulative Poisson probabilities, plus distribution statistics.

The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, when events occur independently at a constant average rate. Classic examples: emergency room arrivals per hour, defects per square meter of fabric, mutations per DNA sequence, customer service calls per minute, or earthquakes per year in a region. The Poisson distribution is named after Siméon Denis Poisson and is fundamental to queuing theory, reliability engineering, and operations research.

This calculator returns Poisson probabilities given a rate parameter λ (lambda) and number of events k. The mean and variance both equal λ — a defining characteristic of Poisson distributions. For most rare events (small probability per trial, large number of trials), Poisson is an excellent approximation to the binomial distribution.

The Poisson is particularly useful when you know the average occurrence rate but want to assess specific outcomes: "what's the probability of exactly 8 customer calls in the next hour if average is 5/hour?" or "is observing 12 defects in 100 m² unusual if we expect 4 defects/m²?" The Poisson distribution provides quick, exact answers without needing the underlying number of trials.

Inputs

Average Rate (lambda)

Number of Events (k)

Results

P(X = k)

0.140374

P(X <= k)

0.265026

P(X > k)

0.734974

Mean (lambda)

5.0000

Variance

5.0000

Std Dev

2.2361

Last updated: May 29, 2026

Formula

**Poisson probability mass function:** P(X = k) = (λ^k × e^(-λ)) / k! Where: - **λ** (lambda): average rate of events - **k**: number of events - **e**: Euler's number ≈ 2.71828 **Worked example: λ = 5 customer calls/hour, P(exactly 3 calls)** P(X = 3) = (5^3 × e^(-5)) / 3! = (125 × 0.00674) / 6 = 0.140 So 14% chance of exactly 3 calls in the next hour. **Cumulative probability:** P(X ≤ k) = Σ P(X = i) for i from 0 to k **Distribution statistics:** - **Mean**: μ = λ - **Variance**: σ² = λ - **Standard deviation**: σ = √λ For λ = 5: mean = 5, variance = 5, SD = 2.24. **Requirements for Poisson:** 1. Events occur independently. 2. Average rate is constant. 3. Two events can't occur simultaneously. 4. Probability of an event is proportional to interval length. **When to use Poisson:** - Counting events in fixed interval. - Average rate known. - Events independent. - No upper bound on count. - Rate per unit (time, space, etc.). **Common applications:** | Field | Application | |---|---| | Operations | Customer arrivals, queue length | | Healthcare | Disease cases per population | | Engineering | Defects per unit | | Insurance | Claims per period | | Telecom | Calls per period | | Biology | Mutations per DNA sequence | | Physics | Radioactive decays per time | | Seismology | Earthquakes per region/year | | Sports | Goals per soccer game | **Examples with typical λ values:** - **Customer calls per hour**: λ = 5-30 typical. - **Manufacturing defects per shift**: λ = 1-5. - **Patient ER arrivals per hour**: λ = 4-20. - **Email spam per hour**: λ = 50-200. - **Webpage visits per minute**: highly variable. **Approximations:** | Distribution | Approximated by Poisson when | |---|---| | Binomial | n large, p small, np reasonable (np < 10) | | Normal | λ large (typically λ ≥ 20) | **Normal approximation:** When λ is large (≥20), Poisson ≈ Normal(λ, λ). **Poisson process:** A Poisson process generates events at constant rate λ over time. Important properties: - Time between events: exponentially distributed. - Number of events in interval of length t: Poisson(λt). - Events in disjoint intervals: independent. **Common probabilities for various λ:** | λ | Mode | SD | P(X=0) | P(X=λ) | P(X<λ) | |---|---|---|---|---|---| | 1 | 1 | 1.0 | 0.368 | 0.368 | 0.368 | | 5 | 5 | 2.24 | 0.007 | 0.176 | 0.616 | | 10 | 10 | 3.16 | 4.5e-5 | 0.125 | 0.583 | | 20 | 20 | 4.47 | 2.1e-9 | 0.089 | 0.557 | | 50 | 50 | 7.07 | ~0 | 0.057 | 0.537 |

How to use this calculator

Enter the average rate (λ) of events.
Enter the number of events (k) you're calculating for.
Calculator returns P(X=k), P(X≤k), and P(X≥k).
For large λ (≥20), normal approximation is accurate.
For small λ (≤5), exact Poisson calculation preferred.
Verify Poisson assumptions: constant rate, independence.

Worked examples

Customer service center calls

**Scenario:** Call center receives 12 calls per hour on average. What's P(exactly 15 calls in next hour)? **Calculation:** λ = 12, k = 15. P(X=15) = (12^15 × e^-12)/15! = (15407… × 6.14e-6)/1307674368000 = 0.072. **Result:** 7.2% probability of exactly 15 calls. The mean is 12, so 15 is somewhat above average but not extreme.

Quality control defects

**Scenario:** Production line produces 4 defects per shift on average. P(no defects in shift)? **Calculation:** λ = 4, k = 0. P(X=0) = (4^0 × e^-4)/0! = 1 × 0.0183 / 1 = 0.018. **Result:** Only 1.8% chance of zero defects. Almost certain to see at least one defect per shift. If true rate is much lower than expected, investigate.

Email spam analysis

**Scenario:** Server averages 50 spam emails per hour. What's P(at most 40 in next hour)? **Calculation:** λ = 50, k = 40. P(X≤40) ≈ 0.086 (using cumulative Poisson, can also use normal approximation since λ≥20). **Result:** Only 8.6% chance of 40 or fewer spam emails. The distribution strongly suggests the volume is consistent — significant variation from 40 would be surprising.

When to use this calculator

**Use Poisson distribution for:**

- **Counting events in continuous intervals**: time, space. - **Rare event modeling**: defects, accidents, mutations. - **Queuing theory**: arrivals, service times. - **Insurance modeling**: claims per period. - **Reliability**: failures per time. - **Network traffic**: packets per second. - **Sports statistics**: goals, hits per game.

**Key requirements:**

- Events independent. - Constant average rate. - Events don't happen simultaneously. - Counts are discrete (0, 1, 2, ...).

**Poisson vs binomial:**

| Aspect | Poisson | Binomial | |---|---|---| | Trials | Continuous interval | Fixed n | | Probability | Rate (λ) | Per trial (p) | | Upper bound | None (theoretically) | n | | Use case | Events in time/space | Fixed trial sequence |

**Poisson approximates binomial when:**

- n is large. - p is small. - np is reasonable (e.g., np < 10). - λ = np.

This makes Poisson easier for rare-event problems with large n.

**Practical interpretations:**

- **P(X = 0)** = e^(-λ): probability of no events. - **P(X ≥ 1)** = 1 - e^(-λ): probability of at least one. - **Mode** = λ rounded down to integer (for integer λ, both λ and λ-1 are modes).

**Poisson process modeling:**

For events occurring continuously: - Average rate λ events per unit time. - Events independent. - Probability of event in small interval Δt is approximately λΔt. - Number of events in time T: Poisson(λT).

**Common error patterns:**

- Using when rate is not constant. - Treating dependent events as independent. - Using for bounded count data. - Computing for negative values. - Ignoring rate dimension (per hour, per square meter, etc.).

**Software:**

- **Excel**: POISSON.DIST(k, λ, FALSE) for exact; TRUE for cumulative. - **R**: dpois(k, λ), ppois(k, λ). - **Python (scipy.stats)**: poisson.pmf(k, λ), poisson.cdf(k, λ). - **SPSS**: COMPUTE WITH PDF.POISSON.

**Real-world considerations:**

- Many "Poisson" processes have time-varying rates (peak hours, seasonal). - For varying rate λ(t): use non-homogeneous Poisson. - For overdispersion (variance > mean): negative binomial may be better. - For underdispersion: binomial may fit better.

**Practical Poisson stats:**

| λ | μ | SD | 95% range | |---|---|---|---| | 5 | 5 | 2.24 | 0-10 | | 10 | 10 | 3.16 | 4-16 | | 25 | 25 | 5.00 | 15-35 | | 100 | 100 | 10.0 | 80-120 |

Common mistakes to avoid

Using when rate is not constant. Time-varying rates need adjustment.
Treating dependent events as independent. Verify independence.
Confusing Poisson with binomial. Poisson is for events in interval; binomial for fixed trials.
Using when counts have upper bound. Use binomial or hypergeometric.
Ignoring rate dimension. λ has units (events per unit time/space).
Computing P for non-integer k. Poisson is discrete.
Forgetting that variance equals mean. Verify with data.

Frequently Asked Questions

Sources & further reading

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