The Poisson distribution is a vital concept in probability theory, particularly useful for modeling the frequency of events occurring in a fixed interval of time or space.

The Basics of Poisson Distribution

The Poisson distribution helps us understand how likely it is for a certain number of events to happen in a fixed interval, given the average number of times these events usually occur. It's useful when we're looking at events that happen independently and at a constant average rate.

Formula

The probability of observing exactly $k$ events is given by:

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

• $e$ is the base of the natural logarithm (about 2.71828).
• $k$ is the number of events we're interested in.
• $\lambda$ is the average number of events.
• $k!$ is $k$ factorial, the product of all positive integers up to $k$.

Key Points

• Discrete: It deals with counts of events, so $k$ can be 0, 1, 2, and so on.
• Versatile: Used in various fields for analyzing rare events over time or space.

Application Steps

1. Identify $\lambda$ : Find the average event rate.

2. Determine $k$: Decide the specific number of events you're interested in.

3. Calculate Probability: Use the formula to calculate the likelihood of (k) 3. events.

Examples

Example 1: Bookstore Customers

Suppose a bookstore averages 3 customers per hour. What is the probability exactly 5 customers arrive in an hour?

Solution:

• Average Rate $(\lambda)$: 3 customers/hour.
• Interested in $(k)$: 5 customers.
• Probability Calculation: Using the formula with $\lambda = 3$ and $k = 5$, we find the probability is about 10.08%.

Example 2: Network Faults

In a network operation centre, an average of 2 faults occur per day. What is the probability that there will be no faults in a day?

Solution:

• Average Rate $(\lambda)$: 2 faults/day.
• Interested in $(k)$: 0 faults.
• Probability Calculation: Using the formula with $\lambda = 2$ and $k = 0$, we find the probability is about 13.53%.