The Poisson distribution is a statistical tool crucial in predicting the probability of a given number of events occurring within a defined period. It's particularly useful when events happen independently and at a constant average rate.
Understanding the Poisson Distribution
Basics
The Poisson distribution shows how likely a number of events will happen in a set period of time or space, assuming these events occur with a fixed average rate and independently of the time since the last event.
Key Points
- Independence: Events don't affect each other.
- Constant Rate: Events happen at a steady average rate.
- Formula: The chance of seeing events is , where is the average number of events, and is the event count.
When to Use
- For random, independent events happen at a constant rate.
- Ideal for rare events over many trials.
Examples
Example 1: Helpdesk Calls
- Scenario: A helpdesk gets 5 calls per hour on average.
- Question: What's the chance of exactly 7 calls in an hour?
- Given: .
- Formula:
- Steps:
- (since )
- Thus,
- Result: The probability of exactly 7 calls is about 10.44%.
Example 2: Radioactive Decay
- Scenario: A substance decays at a rate of 20 particles per minute.
- Question: What's the probability of 25 decays in one minute?
- Given: .
- Formula:
- Steps:
- is a large number, about
- Thus,
- Result: The probability of exactly 25 decays occurring in one minute is approximately 4.46%.