Calls arrive at a rate of λ = 3 per hour. Find P(X = 0).

Poisson Distribution: Formula & Examples

01 — What Is the Poisson Distribution?

The Poisson distribution (named after French mathematician Siméon Denis Poisson, 1837) models the number of times an event occurs in a fixed interval of time or space, under three conditions:

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Three Conditions for a Poisson Process
1. Events occur independently — one event does not make another more or less likely.
2. Events occur at a constant average rate λ (lambda) per interval.
3. Two events cannot occur at exactly the same instant (events are singular).

Real-world examples that satisfy these conditions: the number of phone calls arriving at a call centre per hour; the number of particles emitted by a radioactive source per second; the number of potholes per kilometre of road; the number of typos per page in a book; the number of goals in a football match.

The Poisson distribution is a discrete distribution, X can take values 0, 1, 2, 3, … with no upper bound. The single parameter λ (lambda) is both the mean and the variance of the distribution, a remarkable property we prove below.

02 — The Probability Mass Function (PMF)

If X ~ Poisson(λ), the probability of observing exactly k events is:

Poisson PMF — Core Formula

P(X = k) = (λᵏ · e⁻λ) / k! for k = 0, 1, 2, 3, …

where: λ > 0 = average rate (events per interval) · e ≈ 2.71828 · k! = k factorial

Read: "the probability of exactly k events equals lambda-to-the-k, times e-to-the-minus-lambda, divided by k-factorial."

Verifying the PMF Sums to 1

A legitimate PMF must satisfy Σ P(X = k) = 1 for k = 0, 1, 2, … Substituting the formula:

Σ (λᵏ e⁻λ)/k! = e⁻λ · Σ λᵏ/k! = e⁻λ · eλ = e⁰ = 1 ✓

The last step uses the Maclaurin series for eˣ: eˣ = Σ xᵏ/k!. See Maclaurin Series for the full derivation.

03 — Mean and Variance: Both Equal λ

Mean & Variance of the Poisson Distribution

E[X] = λ Var(X) = λ SD(X) = √λ

The fact that mean = variance is the single most important property of the Poisson distribution — and the quickest way to check if Poisson is appropriate for a dataset.

Deriving E[X] = λ

E[X] = Σ k · (λᵏ e⁻λ) / k!
     = Σ (λᵏ e⁻λ) / (k−1)!   [cancel k with k!]
     = λ e⁻λ · Σ λʲ / j!      [substitute j = k−1]
     = λ e⁻λ · eλ = λ

The variance derivation uses E[X²] = E[X(X−1)] + E[X], showing that E[X(X−1)] = λ², hence Var(X) = E[X²] − (E[X])² = λ² + λ − λ² = λ.

04 — Interactive Distribution Chart

Poisson PMF: Drag the slider to change λ

λ (rate parameter): 3 E[X] = 3 | Var(X) = 3 | SD = 1.73

The distribution shifts right and spreads out as λ increases. For large λ (≥ 10), the shape approximates a normal bell curve.

05 — Probability Table for Selected λ Values

Quick reference: P(X = k) rounded to 4 decimal places:

k	λ = 1	λ = 2	λ = 3	λ = 4	λ = 5
0	0.3679	0.1353	0.0498	0.0183	0.0067
1	0.3679	0.2707	0.1494	0.0733	0.0337
2	0.1839	0.2707	0.2240	0.1465	0.0842
3	0.0613	0.1804	0.2240	0.1954	0.1404
4	0.0153	0.0902	0.1680	0.1954	0.1755
5	0.0031	0.0361	0.1008	0.1563	0.1755
6	0.0005	0.0120	0.0504	0.1042	0.1462
7	0.0001	0.0034	0.0216	0.0595	0.1044
8	0.0000	0.0009	0.0081	0.0298	0.0653

06 — Poisson Approximation to the Binomial

When n is large and p is small, computing the binomial PMF B(n, p) becomes unwieldy. The Poisson distribution provides an excellent approximation:

Poisson Approximation Rule

If X ~ B(n, p) with n large and p small,
then X ≈ Poisson(λ) where λ = np

Guideline: use this approximation when n ≥ 50 and p ≤ 0.1 (or equivalently np ≤ 5). The approximation improves as n → ∞ and p → 0 while np remains constant.

Why Does This Work?

The binomial PMF is C(n,k) pᵏ (1−p)ⁿ⁻ᵏ. Substituting p = λ/n and taking the limit as n → ∞:

C(n,k) · (λ/n)ᵏ · (1−λ/n)ⁿ⁻ᵏ
→ (nᵏ/k!) · (λᵏ/nᵏ) · e⁻λ = λᵏ e⁻λ / k!

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When to use which: Use the Binomial distribution when n is small or p is not small. Switch to Poisson when n ≥ 50, p ≤ 0.1, and you want a simpler calculation.

07 — Cumulative and "At Least" Probabilities

Many problems ask for P(X ≤ k) or P(X ≥ k) rather than exactly k. The strategies are:

Cumulative Poisson Probabilities

P(X ≤ k) = Σⱼ₌₀ᵏ P(X = j)

P(X ≥ k) = 1 − P(X ≤ k−1) [complement rule]

P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a−1)

Always build cumulative sums from k = 0 upward. For P(X ≥ 1), the complement P(X = 0) = e⁻λ is especially clean.

08 — Ten Worked Examples

EX 01 Basic PMF — phone calls per hour

Calls arrive at a switchboard at an average rate of λ = 4 per hour. Find the probability that exactly 2 calls arrive in a given hour.

Identify: X ~ Poisson(4), k = 2.

Apply the formula: P(X=2) = (4² · e⁻⁴) / 2!

= (16 × 0.018316) / 2 = 0.29306 / 2

P(X = 2) ≈ 0.1465 — about 14.65% chance.

EX 02 P(X = 0) — the zero-event probability

A hospital records on average 3 accidents per week. What is the probability of zero accidents in a week?

X ~ Poisson(3), k = 0.

P(X=0) = (3⁰ · e⁻³) / 0! = 1 · e⁻³ / 1 = e⁻³

e⁻³ ≈ 0.0498

P(X = 0) ≈ 0.0498 — only a 5% chance of a zero-accident week.

EX 03 Cumulative — P(X ≤ 2)

With λ = 3, find P(X ≤ 2).

P(X=0) = e⁻³ ≈ 0.0498

P(X=1) = 3e⁻³ ≈ 0.1494

P(X=2) = (9/2)e⁻³ ≈ 0.2240

P(X ≤ 2) = 0.0498 + 0.1494 + 0.2240

P(X ≤ 2) ≈ 0.4232

EX 04 At least one — P(X ≥ 1)

A Geiger counter detects radioactive particles at a rate of λ = 2 per minute. Find P(X ≥ 1) in one minute.

Use the complement: P(X ≥ 1) = 1 − P(X = 0)

P(X = 0) = e⁻² ≈ 0.1353

P(X ≥ 1) = 1 − 0.1353

P(X ≥ 1) ≈ 0.8647 — an 86.5% chance of detecting at least one particle.

EX 05 Rescaling the interval — from hourly to 30-minute rate

Emails arrive at a rate of 6 per hour. What is the probability of receiving exactly 4 emails in 30 minutes?

Scale λ to the new interval: 30 min = ½ hour, so λ₃₀ = 6 × ½ = 3

X ~ Poisson(3), k = 4: P(X=4) = (3⁴ · e⁻³) / 4!

= (81 × 0.0498) / 24 = 4.0338 / 24

P(X = 4) ≈ 0.1680 — about 16.8% chance.

EX 06 Poisson approximation to binomial

A factory produces 1,000 bolts per day. Each bolt has a 0.005 (0.5%) probability of being defective. Find the approximate probability that exactly 3 bolts are defective, using the Poisson approximation.

Check conditions: n = 1000 (large), p = 0.005 (small) ✓. Use Poisson with λ = np = 1000 × 0.005 = 5

P(X=3) = (5³ · e⁻⁵) / 3! = (125 × 0.006738) / 6

= 0.84224 / 6 ≈ 0.1404

P(X = 3) ≈ 0.1404 — exact binomial gives 0.1402, confirming the approximation is excellent.

EX 07 Mean and standard deviation

A call centre receives on average 9 calls per hour. State the mean, variance, and standard deviation of the number of calls per hour.

X ~ Poisson(9). E[X] = λ = 9

Var(X) = λ = 9

SD(X) = √9 = 3

Mean = 9, Variance = 9, SD = 3 — mean = variance is the Poisson hallmark.

EX 08 P(X > 2) using complement

With λ = 1.5, find P(X > 2).

P(X > 2) = 1 − P(X ≤ 2)

P(0) = e⁻¹·⁵ ≈ 0.2231, P(1) = 1.5e⁻¹·⁵ ≈ 0.3347, P(2) = (2.25/2)e⁻¹·⁵ ≈ 0.2510

P(X ≤ 2) ≈ 0.2231 + 0.3347 + 0.2510 = 0.8088

P(X > 2) = 1 − 0.8088

P(X > 2) ≈ 0.1912

EX 09 Finding λ from a given probability

The probability of zero events in an interval is 0.0821. Find λ.

P(X = 0) = e⁻λ = 0.0821

Take natural log: −λ = ln(0.0821) ≈ −2.5

λ = 2.5

λ = 2.5 events per interval.

EX 10 Range probability — P(2 ≤ X ≤ 4)

With λ = 2.5, find P(2 ≤ X ≤ 4).

P(X=2) = (2.5² · e⁻²·⁵)/2! = (6.25 × 0.0821)/2 ≈ 0.2565

P(X=3) = (2.5³ · e⁻²·⁵)/3! = (15.625 × 0.0821)/6 ≈ 0.2138

P(X=4) = (2.5⁴ · e⁻²·⁵)/4! = (39.0625 × 0.0821)/24 ≈ 0.1336

P(2 ≤ X ≤ 4) = 0.2565 + 0.2138 + 0.1336

P(2 ≤ X ≤ 4) ≈ 0.6039 — about a 60% chance.

09 — Common Mistakes

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Mistake 1 — Wrong interval: If λ is given per hour but the problem asks about 15 minutes, you must rescale: λ₁₅ = λ × (15/60). Forgetting this is the most common Poisson error.

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Mistake 2 — Using Poisson when independence fails: If events cluster together (e.g., earthquakes triggering aftershocks, or a busy period causing correlated arrivals), the Poisson model is inappropriate.

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Mistake 3 — Forgetting 0! = 1: For k = 0, the formula gives λ⁰ e⁻λ / 0! = e⁻λ, not zero. Students sometimes write 0! = 0 by analogy with 0 × anything = 0. By convention, 0! = 1.

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Mistake 4 — Confusing P(X = k) with P(X ≤ k): "At most 3 events" means P(X ≤ 3) = P(0) + P(1) + P(2) + P(3). "Exactly 3 events" means only P(3). Read the question carefully.

🔗 Related Topics — Probability & Statistics

Basic Probability Conditional Probability Bayes' Theorem Discrete Distributions Binomial Distribution Normal Distribution Continuous Distributions Expected Value & Variance Hypothesis Testing Maclaurin Series All Probability Topics

The Poisson Distribution
Modelling Rare Events in Time and Space

01 — What Is the Poisson Distribution?

02 — The Probability Mass Function (PMF)

Verifying the PMF Sums to 1

03 — Mean and Variance: Both Equal λ

Deriving E[X] = λ

04 — Interactive Distribution Chart

05 — Probability Table for Selected λ Values

06 — Poisson Approximation to the Binomial

Why Does This Work?

07 — Cumulative and "At Least" Probabilities

08 — Ten Worked Examples

09 — Common Mistakes

🔗 Related Topics — Probability & Statistics

10 — Quiz: Poisson Distribution (10 Questions)