Poisson Distribution Table

Poisson Distribution Table

The Poisson distribution models the number of events occurring in a fixed interval of time or space, given a known average rate (λ).

Poisson Probability Formula

The probability of observing exactly k events is:

P(X = k) = (e^(-λ) × λ^k) / k!

Where:

• λ (lambda) = average rate (mean number of events)
• k = number of events
• e ≈ 2.71828

Cumulative Probabilities P(X ≤ k)

λ = 0.5 to 2.5

λ = 3.0 to 5.0

λ = 6.0 to 10.0

λ = 12 to 20

Individual Probabilities P(X = k)

λ = 5.0

How to Calculate Probabilities

Individual Probability: P(X = k)

$P(X = k) = P(X \leq k) - P(X \leq k-1)$

“At least” Probability: P(X ≥ k)

$P(X \geq k) = 1 - P(X \leq k-1)$

Range: P(a ≤ X ≤ b)

$P(a \leq X \leq b) = P(X \leq b) - P(X \leq a-1)$

When to Use Poisson Distribution

✓ Events occur independently
✓ The rate (λ) is constant
✓ Two events cannot occur simultaneously
✓ Counting events in a fixed interval

Common Applications

Poisson Mean and Variance

Note: For Poisson, mean = variance = λ

Example Problems

Example 1: Exact Probability

Problem: A call center receives an average of 4 calls per hour. What is P(exactly 3 calls)?

Solution (λ = 4):

• P(X ≤ 3) = 0.4335
• P(X ≤ 2) = 0.2381
• P(X = 3) = 0.4335 - 0.2381 = 0.1954

Example 2: At Most

Problem: What is P(X ≤ 2) when λ = 4?

Solution: Read directly from table

• P(X ≤ 2) = 0.2381

Example 3: At Least

Problem: What is P(X ≥ 5) when λ = 4?

Solution:

• P(X ≥ 5) = 1 - P(X ≤ 4)
• P(X ≥ 5) = 1 - 0.6288 = 0.3712

Poisson Approximation to Binomial

When n is large (n ≥ 20) and p is small (p ≤ 0.05):

$\text{Binomial}(n, p) \approx \text{Poisson}(\lambda = np)$

Related Resources

• Poisson Distribution Lesson - Complete guide
• Binomial Table - For fixed trials
• Probability Calculator - Calculate probabilities
• Exponential Distribution - Time between events