Updated Probability & Statistics Tool

Poisson Distribution Calculator

Compute Poisson probabilities for event counts in a fixed interval. Get P(X = k), P(X ≤ k), P(X ≥ k), mean, variance, standard deviation, Poisson probability tables, inverse Poisson and normal approximation in one place.

PMF & CDF Tail Probabilities Inverse Poisson Normal Approximation

Calculate Poisson Probabilities for Counts per Interval

Use this Poisson Distribution Calculator to model the number of events in a fixed interval when events occur independently at a constant average rate. Enter the rate parameter λ (lambda) and the number of events k to compute the probability mass function, cumulative probabilities, tail probabilities and distribution summary statistics. You can also generate Poisson tables, solve simple inverse Poisson problems and compare with the normal approximation.

λ (lambda) is the expected number of events per interval (must be positive). k is the number of events (a non-negative integer). This calculator focuses on numeric accuracy and clear interpretation of probabilities.

For the “between” option, probabilities are computed by summing the Poisson PMF from k1 to k2 inclusive. All probabilities are rounded for display but computed internally in double precision.

The Poisson table lists probabilities P(X = k) and cumulative probabilities P(X ≤ k) from k = 0 up to your chosen maximum. This is useful for quick lookups and sanity checks for homework or reports.

Inverse Poisson problems are common in service-level or reliability settings (for example, “how many calls can we handle so that the probability of more than k calls is under 5%?”). This tab searches integer k values up to your chosen limit using cumulative Poisson probabilities.

For large λ, the Poisson distribution can be approximated by a normal distribution with mean λ and variance λ. This tab compares the exact Poisson CDF P(X ≤ k) with the normal approximation Φ((k + 0.5 − λ) / √λ), where Φ is the standard normal CDF.

Poisson Distribution Calculator – Complete Guide to Counts per Interval

The Poisson Distribution Calculator on MyTimeCalculator is a practical tool for modeling counts of events that occur randomly in time or space, such as incoming calls per minute, defects per meter, arrivals per hour or emails per day. When events occur independently and with a constant average rate, the Poisson distribution provides a simple and powerful model for the number of events in a fixed interval.

This calculator brings together the key Poisson quantities in one interface: the probability mass function P(X = k), cumulative probability P(X ≤ k), tail probability P(X ≥ k), mean, variance, standard deviation, Poisson tables for a range of k, inverse Poisson calculations and a normal approximation for large rates.

1. The Poisson Distribution at a Glance

A random variable X is said to follow a Poisson distribution with parameter λ > 0 if

P(X = k) = e−λ λk / k!,   for k = 0, 1, 2, …

Here, λ represents the expected number of events per interval. For example, λ = 4 could represent an average of four calls per hour at a service desk. The Poisson distribution is discrete, defined only for non-negative integer values of k.

A key feature of the Poisson distribution is that its mean and variance are equal:

E[X] = λ,    Var(X) = λ,    σ = √λ.

The calculator uses these relationships to report mean, variance and standard deviation as soon as you specify λ.

2. Computing PMF, CDF and Tail Probabilities

The main tab of the calculator focuses on three core probabilities:

  • PMF: P(X = k), the probability of exactly k events.
  • CDF: P(X ≤ k), the probability of k or fewer events.
  • Tail: P(X ≥ k), the probability of at least k events.

For Poisson distributions, P(X ≤ k) is computed by summing the PMF from 0 up to k. The tail probability

P(X ≥ k) = 1 − P(X ≤ k − 1)

is obtained from the complement of the CDF. The calculator evaluates these numerically using a stable iterative formula that builds probabilities from k = 0 upward to minimize rounding error.

3. Between Probabilities P(k₁ ≤ X ≤ k₂)

Many practical questions involve a range of counts, such as “What is the probability of having between two and six calls inclusive?” For this, the calculator provides the between option:

P(k₁ ≤ X ≤ k₂) = Σk=k₁k₂ P(X = k).

Internally, the calculator sums the Poisson PMF from k = k₁ to k = k₂ and reports the resulting probability, while still showing the single-point PMF at k, the CDF at k and the tail probability at k for reference.

4. Poisson Distribution Summary Statistics

Once λ is known, the primary summary statistics follow immediately:

  • Mean E[X] = λ: the average number of events per interval.
  • Variance Var(X) = λ: the variability of the counts around the mean.
  • Standard deviation σ = √λ: the typical deviation from the mean.

The calculator displays these values on the main tab so that you can interpret probabilities in the context of how concentrated or spread the counts are around their expected value.

5. Poisson Probability Tables

Tables of Poisson probabilities are still widely used for classroom exercises and quick planning. On the Poisson table tab, you specify λ and a maximum k, and the calculator generates a table with:

  • k, the number of events.
  • P(X = k), the probability of exactly k events.
  • P(X ≤ k), the cumulative probability up to k.

This table is useful for sanity checks, visualizing how probability mass is distributed across different values of k, or building custom graphics and reports using exported values.

6. Inverse Poisson Problems

In planning and risk management, a common question is “What threshold k achieves a certain probability level?”. Examples include:

  • Choosing k so that P(X ≤ k) ≥ 0.95 (a 95% service level).
  • Choosing k so that P(X ≥ k) ≤ 0.01 (a rare-event tolerance).

The inverse Poisson tab allows you to specify λ, a target probability p and whether you want a cumulative or tail condition. The calculator then searches integer k values up to a user-defined limit to find the smallest k that meets the requirement and reports the corresponding probability.

7. Normal Approximation to Poisson

For large λ, the Poisson distribution is well-approximated by a normal distribution:

X ≈ N(λ, λ).

Since the Poisson distribution is discrete and the normal distribution is continuous, a continuity correction is typically applied. For example, to approximate P(X ≤ k), one uses:

P(X ≤ k) ≈ Φ( (k + 0.5 − λ) / √λ )

where Φ is the standard normal CDF. The normal approximation tab computes both the exact Poisson CDF and the corresponding normal approximation, along with the z-score, so that you can see how accurate the approximation is for your specific λ and k.

8. Typical Applications of the Poisson Distribution

  • Queueing and call centers: arrivals of customers or calls per unit time.
  • Quality control: defects per item, meter or batch.
  • Reliability: failures of components over a time horizon.
  • Traffic and networks: packets per second, hits per second on a server.
  • Epidemiology: rare disease cases within a fixed population and time window.

In each of these contexts, λ encapsulates the underlying rate of occurrence. By choosing an appropriate interval (hour, day, month, unit length and so on) and estimating λ from data, the Poisson model can inform capacity planning, risk assessment and performance guarantees.

9. How to Use the Poisson Distribution Calculator Effectively

  1. Identify the interval of interest and estimate λ as the average number of events per interval.
  2. Enter λ and the value of k you care about (or a range [k₁, k₂]).
  3. Use the main tab to compute P(X = k), P(X ≤ k) and P(X ≥ k).
  4. Switch to the Poisson table tab to see how probability is distributed across different k values.
  5. Use the inverse Poisson tab for service-level style questions where you need to solve for k.
  6. For large λ, compare exact Poisson probabilities with the normal approximation to understand accuracy.

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Poisson Distribution Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about Poisson probabilities, when the model applies and how to interpret the calculator outputs.

The Poisson distribution is appropriate when you count the number of events in a fixed interval and the events occur independently, with a constant average rate and with a very small chance of more than one event occurring at the exact same instant. Classic examples include calls to a contact center, website hits per minute, defects per manufactured item and arrivals at a service desk.

λ (lambda) is the expected number of events per chosen interval. If you observe on average eight calls per hour, you can model the number of calls in an hour with a Poisson distribution where λ = 8. Changing the interval changes λ, so it is important to be clear about the time or space unit you are using.

In the Poisson model, events occur independently at rate λ. The mathematical structure of this process implies that the expected value of the count in one interval is λ and the variance, which measures dispersion around the mean, also turns out to be λ. This is one of the distinctive fingerprints of the Poisson distribution and is often used as a quick check of model suitability in real data sets.

As a rule of thumb, the normal approximation becomes reasonably accurate when λ is moderately large (for example, λ ≥ 10–15) and you apply a continuity correction. For small λ, the Poisson distribution is noticeably skewed and discrete effects are important, so the exact Poisson probabilities from this calculator are preferred over a normal approximation.

In real data, it is common to see overdispersion, where the variance of the counts is larger than the mean. This can indicate that the simple Poisson model is not adequate, perhaps due to clustering, time- varying rates or unobserved heterogeneity. In such cases, more flexible models such as the negative binomial distribution may be more appropriate, but the Poisson calculator can still provide a useful baseline comparison.

Yes. The Poisson Distribution Calculator is well suited for checking numeric answers on homework, exam preparation and lab reports. You can replicate the parameters given in the problem, compare your manual calculations with the computed PMF, CDF or tail probabilities, and use the table and inverse tabs to explore how different thresholds affect risk or service levels.

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