Poisson Distribution

Overview

The Poisson distribution models the number of events occurring in a fixed interval of time or space, when events happen at a constant average rate and independently.

Conditions

• Events occur independently
• Events occur at a constant average rate
• Two events cannot occur at exactly the same instant
• The probability of an event is proportional to the interval length

Parameter

$\lambda$ (lambda): The average number of events per interval

Probability Mass Function

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

Where:

• $k = 0, 1, 2, 3, \ldots$ (any non-negative integer)
• $e \approx 2.71828$ (Euler's number)

Key Formulas

Mean

$$\mu = E(X) = \lambda$$

Variance

$$\sigma^2 = \text{Var}(X) = \lambda$$

Note: Mean equals variance—a unique property of Poisson!

Standard Deviation

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

Notation

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

or

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

Changing the Interval

If $\lambda$ is the rate per unit, then for $t$ units:

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

Applications

Context	$\lambda$ represents
Customer arrivals	Avg arrivals per hour
Website visits	Avg visits per minute
Typos per page	Avg errors per page
Accidents	Avg accidents per month
Calls to call center	Avg calls per hour

Poisson as Binomial Limit

When $n$ is large and $p$ is small, with $\lambda = np$:

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

Rule of thumb: Use when $n \geq 20$ and $p \leq 0.05$

Examples

Example 1: Basic Calculation

A call center receives an average of 4 calls per hour. Find $P(\text{exactly 6 calls in an hour})$.

$$\lambda = 4, \quad k = 6$$ $$P(X = 6) = \frac{e^{-4} \cdot 4^6}{6!} = \frac{0.0183 \times 4096}{720} = 0.1042$$

Example 2: Changing Time Interval

If $\lambda = 4$ calls per hour, find $P(\text{2 or fewer calls in 30 minutes})$.

For 30 min: $\lambda = 4 \times 0.5 = 2$

$$P(X \leq 2) = P(0) + P(1) + P(2)$$ $$P(0) = \frac{e^{-2} \cdot 2^0}{0!} = 0.1353$$ $$P(1) = \frac{e^{-2} \cdot 2^1}{1!} = 0.2707$$ $$P(2) = \frac{e^{-2} \cdot 2^2}{2!} = 0.2707$$ $$P(X \leq 2) = 0.1353 + 0.2707 + 0.2707 = 0.6767$$

Example 3: At Least One Event

A website averages 3 errors per day. $P(\text{at least one error})$?

$$P(X \geq 1) = 1 - P(X = 0) = 1 - \frac{e^{-3} \cdot 3^0}{0!} = 1 - e^{-3} = 1 - 0.0498 = 0.9502$$

Example 4: Rare Events (Binomial → Poisson)

A large batch of 1000 items has 0.5% defective. $P(\text{exactly 3 defective})$?

$$n = 1000, \quad p = 0.005, \quad \lambda = np = 5$$ $$P(X = 3) = \frac{e^{-5} \cdot 5^3}{3!} = \frac{0.00674 \times 125}{6} = 0.1404$$

Poisson Table Values

Common $P(X = k)$ for selected $\lambda$:

$k$	$\lambda=1$	$\lambda=2$	$\lambda=3$	$\lambda=5$
0	0.368	0.135	0.050	0.007
1	0.368	0.271	0.149	0.034
2	0.184	0.271	0.224	0.084
3	0.061	0.180	0.224	0.140
4	0.015	0.090	0.168	0.175
5	0.003	0.036	0.101	0.175

Properties

• Mode $\approx \lambda$ (exactly $\lambda$ if $\lambda$ is integer, $\lfloor\lambda\rfloor$ otherwise)
• Skewness $= 1/\sqrt{\lambda}$ (right-skewed, approaches symmetric as $\lambda$ increases)
• Sum of independent Poisson: $\text{Pois}(\lambda_1) + \text{Pois}(\lambda_2) = \text{Pois}(\lambda_1 + \lambda_2)$