Poisson Distribution

Calculate probabilities for the number of events occurring in a fixed interval of time or space

Poisson Distribution Calculator

Distribution Parameter

Average number of events in the given interval (λ > 0)

Calculation Type

Calculate P(X = k), the probability of observing exactly k events

Display Options

Poisson Distribution Visualization

λ = 3

Probability Table

k P(X = k) P(X ≤ k)

Poisson Distribution Overview

The Poisson distribution models the number of events occurring in a fixed interval of time or space, assuming these events happen with a known constant mean rate and independently of each other.

Probability Mass Function (PMF)

\[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]

where λ is the average number of events in the interval and k is the number of events.

Cumulative Distribution Function (CDF)

\[ P(X \leq k) = \sum_{i=0}^{k} \frac{e^{-\lambda} \lambda^i}{i!} = e^{-\lambda} \sum_{i=0}^{k} \frac{\lambda^i}{i!} \]

Expected Value and Variance

\[ E[X] = Var(X) = \lambda \]

Key Properties

  • Mean and variance are both equal to λ
  • As λ increases, the distribution becomes more symmetric and approaches a normal distribution
  • The sum of independent Poisson random variables is also Poisson distributed
  • The Poisson distribution can be derived as the limit of a binomial distribution as n → ∞ and p → 0 while np = λ remains constant
  • Skewness = 1/√λ and Kurtosis = 3 + 1/λ

Understanding the Poisson Distribution

Definition & Characteristics

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, assuming these events occur with a known constant mean rate and independently of the time since the last event.

Requirements for a Poisson Process

  1. Independent events: The occurrence of one event does not affect the probability of another event occurring.
  2. Events occur one at a time: Two or more events cannot occur at exactly the same time.
  3. Constant rate: The rate at which events occur is constant throughout the interval.
  4. Rare events: The probability of an event in a very small interval is proportional to the length of the interval.

Mathematical Formulation

A random variable X follows a Poisson distribution with parameter λ (lambda), written as X ~ Poisson(λ), if:

  • X represents the number of events occurring in a fixed interval
  • λ represents the average number of events in that interval
  • Events occur independently of each other
  • The probability of observing k events is given by the PMF: \( P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \)

Relationship with Binomial Distribution

The Poisson distribution can be derived as a limiting case of the binomial distribution where:

  • The number of trials n becomes very large (n → ∞)
  • The probability of success p becomes very small (p → 0)
  • The product np = λ remains constant

Properties & Applications

Key Properties

Expected Value (Mean)
E[X] = λ
The average number of events in the given interval
Variance
Var(X) = λ
The expected squared deviation from the mean, equal to λ
Standard Deviation
σ = √λ
Measure of the dispersion of the distribution
Skewness
1/√λ
Measure of the asymmetry of the distribution
Mode
⌊λ⌋ (when λ is not an integer)
The most likely number of events

Additivity Property

If X ~ Poisson(λ₁) and Y ~ Poisson(λ₂) are independent, then X + Y ~ Poisson(λ₁ + λ₂). This property makes the Poisson distribution useful for modeling composite processes.

Normal Approximation

For large values of λ (typically λ > 10), the Poisson distribution can be approximated by a normal distribution with mean λ and variance λ.

Real-World Applications

  • Call centers: Number of calls received per hour
  • Quality control: Number of defects in a product or manufacturing process
  • Biology: Number of mutations in a DNA sequence
  • Insurance: Number of claims in a time period
  • Telecommunications: Number of data packets arriving at a router
  • Astronomy: Number of stars in a region of space
  • Radioactivity: Number of radioactive particles detected in a time interval
  • Epidemiology: Number of disease cases in a population

Example Problems

Example 1: Call Center

A call center receives an average of 12 calls per hour. What is the probability of receiving exactly 8 calls in a particular hour?

This is a Poisson distribution problem where:

  • λ = 12 (average number of calls per hour)
  • k = 8 (we want the probability of exactly 8 calls)
\begin{align} P(X = 8) &= \frac{e^{-\lambda} \lambda^k}{k!} \\ &= \frac{e^{-12} \cdot 12^8}{8!} \\ &= \frac{e^{-12} \cdot 429,981,696}{40,320} \\ &= e^{-12} \cdot 10,663.98 \\ &\approx 0.0752 \end{align}

The probability of receiving exactly 8 calls in an hour is about 0.0752 or 7.52%.

Example 2: Manufacturing Defects

A manufacturing process produces circuit boards with an average of 2.5 defects per board. What is the probability that a randomly selected board has at most 1 defect?

This is a Poisson distribution problem where:

  • λ = 2.5 (average number of defects per board)
  • We want P(X ≤ 1), the probability of at most 1 defect
\begin{align} P(X \leq 1) &= P(X = 0) + P(X = 1) \\ &= \frac{e^{-2.5} \cdot 2.5^0}{0!} + \frac{e^{-2.5} \cdot 2.5^1}{1!} \\ &= e^{-2.5} \cdot 1 + e^{-2.5} \cdot 2.5 \\ &= e^{-2.5} \cdot (1 + 2.5) \\ &= e^{-2.5} \cdot 3.5 \\ &\approx 0.0821 \cdot 3.5 \\ &\approx 0.2873 \end{align}

The probability that a randomly selected board has at most 1 defect is about 0.2873 or 28.73%.

Example 3: Radioactive Decay

A Geiger counter detects an average of 15 radioactive particles per minute from a sample. What is the probability of detecting between 10 and 20 particles (inclusive) in a one-minute period?

This is a Poisson distribution problem where:

  • λ = 15 (average number of particles per minute)
  • We want P(10 ≤ X ≤ 20), the probability of detecting between 10 and 20 particles

We can calculate this using the formula P(a ≤ X ≤ b) = P(X ≤ b) - P(X ≤ a-1):

\begin{align} P(10 \leq X \leq 20) &= P(X \leq 20) - P(X \leq 9) \\ \end{align}

Using the Poisson CDF formula or a calculator:

\begin{align} P(X \leq 20) &\approx 0.9245 \\ P(X \leq 9) &\approx 0.0988 \\ P(10 \leq X \leq 20) &= 0.9245 - 0.0988 \\ &= 0.8257 \end{align}

The probability of detecting between 10 and 20 particles in a minute is about 0.8257 or 82.57%.

Example 4: Normal Approximation

A large website averages 500 visitors per hour. Using the normal approximation to the Poisson distribution, estimate the probability that there will be more than 525 visitors in a particular hour.

Since λ = 500 is large, we can approximate the Poisson distribution with a normal distribution:

  • X ~ Poisson(500) can be approximated by X ~ N(500, 500)
  • We want P(X > 525)

Using the Z-score formula:

\begin{align} Z &= \frac{X - \mu}{\sigma} = \frac{525.5 - 500}{\sqrt{500}} \\ &= \frac{25.5}{22.36} \\ &= 1.14 \end{align}

Note: We use 525.5 for continuity correction since we're approximating a discrete distribution with a continuous one.

Using the standard normal table or calculator:

\begin{align} P(X > 525) &= P(Z > 1.14) \\ &= 1 - P(Z \leq 1.14) \\ &= 1 - 0.8729 \\ &= 0.1271 \end{align}

The approximate probability of having more than 525 visitors in an hour is about 0.1271 or 12.71%.

Test Your Knowledge

Quick Quiz: Poisson Distribution

1. A bakery sells an average of 24 loaves of bread per day. What is the probability of selling exactly 20 loaves on a given day?

A. 0.0605
B. 0.0843
C. 0.0758
D. 0.1015

2. For a Poisson distribution with parameter λ = 5, what is the variance?

A. 5
B. √5
C. 25
D. 1/5

3. Which of the following is NOT a characteristic of a Poisson process?

A. Events occur independently
B. The probability of an event is proportional to the length of the interval
C. The rate of events varies significantly over time
D. Two or more events cannot occur at exactly the same time