Poisson Distribution

 

📊 Poisson Distribution 

The Poisson distribution models the number of times an event occurs in a fixed interval (time, space, area, volume, etc.) when events happen:

• independently

• at a constant average rate

• rarely relative to the interval size

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When do we use Poisson?

Use it when counting events like:

• 📞 number of calls arriving at a call center per minute

• 🚗 number of cars passing a signal in 1 minute

• 🧬 mutations in a DNA segment

• 🌧 number of raindrops hitting a sensor per second

• 🏥 number of patients arriving in an hour

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Probability formula

If

$$X \sim \text{Poisson}(\lambda)$$

then

$$P(X=k)=\frac{\lambda^k e^{-\lambda}}{k!}, \quad k=0,1,2,\dots$$

Parameters

Symbol	Meaning
$k$	number of events
$\lambda$	average events per interval (rate)
$e$	Euler's constant ≈ 2.718

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Key properties

Mean and variance

$$E[X]=\lambda$$ $$\text{Var}(X)=\lambda$$

👉 Mean equals variance — a special Poisson feature.

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Standard deviation

$$\sigma=\sqrt{\lambda}$$

Example 1 — Phone calls

A call center receives 4 calls per minute on average.

$$\lambda=4$$

Probability of exactly 2 calls in a minute:

$$P(X=2)=\frac{4^2 e^{-4}}{2!} =\frac{16e^{-4}}{2} =8e^{-4}$$

Since

$$e^{-4}\approx0.0183$$
$$P(X=2)\approx8\times0.0183=0.146$$

✅ About 14.6% chance

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Example 2 — Rare defects

A factory produces 0.5 defects per meter of wire.

$$\lambda=0.5$$

Probability of zero defects in one meter:

$$P(X=0)=e^{-0.5}=0.6065$$

✅ About 60.6% chance

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Example 3 — Cars at a signal

Average 6 cars per minute

$$\lambda=6$$

Probability of at most 1 car:

$$P(X\le1)=P(0)+P(1)$$
$$P(0)=e^{-6}=0.00248$$
$$P(1)=6e^{-6}=0.0149$$
$$P(X\le1)=0.0174$$

✅ Very unlikely (≈1.7%)

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Intuition: Why Poisson works

Imagine dividing time into tiny slots:

• probability of event in each slot is very small

• events independent

This is like a limit of Binomial distribution:

$$\text{Binomial}(n,p),\quad n\to\infty,\; p\to0,\; np=\lambda$$

Then:

$$\text{Binomial} \rightarrow \text{Poisson}$$

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Shape of Poisson distribution

• Small λ → skewed right

• Large λ → becomes symmetric

• Very large λ → approximates Normal distribution

Rule of thumb:

$$\lambda>10 \Rightarrow \text{Normal approx works well}$$

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Additivity property (very useful)

If

$$X_1\sim\text{Poisson}(\lambda_1),\quad X_2\sim\text{Poisson}(\lambda_2)$$

independent, then

$$X_1+X_2\sim\text{Poisson}(\lambda_1+\lambda_2)$$

Example:

• morning calls: λ=3

• afternoon calls: λ=5

Total day:

$$\lambda=8$$

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Connection to waiting times

Poisson counts events.

The time between events follows:

$$\text{Exponential}(\lambda)$$

This link is fundamental in:

• queueing theory

• reliability engineering

• survival analysis

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Real-life interpretation of λ

$$\lambda = \text{rate} \times \text{interval length}$$

Example:

• 2 emails/hour

• in 3 hours:

$$\lambda=2\times3=6$$

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Small Python example

from math import exp, factorial

lam = 4
k = 2

p = (lam**k * exp(-lam)) / factorial(k)
print(p)

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✔️ Summary

Poisson distribution models:

✅ counts of events
✅ independent occurrences
✅ constant average rate
✅ rare events in small intervals

Formula:

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

Mean = Variance = λ