Poisson Distribution 🐟

Named after French mathematician S.D. Poisson. It is a limiting case of Binomial Distribution. It is used for rare events.

When to use? (Conditions) 🛑

1. n is very large (→ ∞).
2. p is very small (→ 0).
3. np = m is finite (Average is constant).

Examples of Rare Events

• Number of mistakes per page in a book.
• Number of accidents on a road per month.
• Number of customers arriving at a counter per minute.
• Number of defective bulbs in a large batch.

The Formula ⚗️

P(r) = (e^-m * m^r) / r!

Where:

• m = Average number of occurrences (Mean).
• r = Number of successes required.
• e = Constant (approx 2.71828).

Properties 📊

[!IMPORTANT] Unique Feature: In Poisson Distribution, Mean = Variance = m.

Example

If average mistakes per page is 2 (m=2). Find prob of 0 mistakes. P(0) = (e^-2 * 2^0) / 0! P(0) = (0.1353 * 1) / 1 = 0.1353.

(Value of e^-2 is taken from tables, usually 0.1353)