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):

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

Frequently Asked Questions

What is the Poisson distribution used for?

The Poisson distribution models the number of times an event occurs in a fixed interval (time, area, volume) when events happen independently at a constant average rate. Examples include number of emails per hour, defects per unit, or customer arrivals per day.

How do I read the Poisson table?

Find your λ (lambda) value in the column headers and your k value in the rows. The table value gives P(X ≤ k) — the probability of observing k or fewer events. For exact probabilities, subtract adjacent cumulative values.

What does λ (lambda) represent?

Lambda (λ) is the average rate — the expected number of events per interval. For example, if a store averages 3 customers per minute, then λ = 3.

How do I find P(X = k) from cumulative probabilities?

Subtract: P(X = k) = P(X ≤ k) − P(X ≤ k−1). For example, if λ = 4: P(X = 3) = P(X ≤ 3) − P(X ≤ 2) = 0.4335 − 0.2381 = 0.1954.

How do I find P(X ≥ k)?

Use the complement: P(X ≥ k) = 1 − P(X ≤ k−1). For example, P(X ≥ 5) when λ = 4: 1 − P(X ≤ 4) = 1 − 0.6288 = 0.3712.

When can I approximate binomial with Poisson?

When n ≥ 20 and p ≤ 0.05 (many trials, small probability), use λ = np. This is useful because the Poisson table is simpler to use than the binomial for large n.

What is the relationship between Poisson and exponential distributions?

If events occur at a Poisson rate λ, then the time between events follows an exponential distribution with parameter λ. Poisson counts events; exponential measures waiting time.

What if my λ value is not in the table?

For λ values between those listed, use the Poisson formula directly: P(X = k) = e^(−λ) × λ^k / k!, or use our Probability Calculator.

Related Resources

• Poisson Distribution Lesson - Complete guide
• Binomial Table - For fixed trials
• Probability Calculator - Calculate probabilities
• Exponential Distribution - Time between events
• Z-Table - Standard normal distribution
• Chi-Square Table - For categorical data tests