**What is Poisson Distribution?**

The Poisson Distribution is a statistical distribution that represents **the number of events occurring in a fixed interval of time or space**.

These events must occur with a known constant mean rate and be independent of the time since the last event.

**Example**

**Imagine a call center where the average number of calls received is 5 per hour.**

The Poisson Distribution can help predict the probability of receiving a certain number of calls **in any given hour**.

**Key Takeaways**

- Used for predicting the number of times an event will happen
**within a specific time frame**. - Ideal for
**rare events**that occur at a constant average rate. **Events are independent**of each other, meaning one event doesn’t affect the probability of another.**The mean and variance**of the Poisson Distribution are both equal to , the average rate of occurrence.- It’s a discrete distribution, meaning the events can only take on
**integer values**like 0, 1, 2, and so on.

**Formula**

The formula for the Poisson Distribution is:

- is the probability of %\quicklatex{color=”#323737″ size=18}k% events in the interval.
- is the average number of events in the interval.
- is approximately equal to 2.71828 (Euler’s number).
- is
**the actual number of successes**that result from the experiment.

**λ Defines the Shape of the Graph**

As increases in the Poisson Distribution, the distribution becomes **more spread out**, and its peak shifts to the right.

When is large, **the distribution begins to resemble a normal distribution**, and thus the Central Limit Theorem can be applied, making it approximate a standard normal distribution.

**Example Calculation**

Using the call center example, if (average 5 calls per hour), what’s the probability of receiving exactly **3 calls in an hour**?

Breaking it down:

- represents the probability of no calls.
- represents the probability of 3 calls.
- adjusts for the number of ways 3 calls can happen.

**Mean and Variance**

For the Poisson Distribution, the mean and variance are both equal to

- Mean =
- Variance =

If a call center receives an average of 5 calls per hour, then both the mean and variance for the number of calls received in an hour using the Poisson Distribution are 5.

**Exercise with Poisson Distribution**

Imagine a bookstore. That bookstore, on average, sells **4 books** **every 2 hours**.

What’s the probability they sell exactly 2 books in the next 2 hours?

**Assumptions**

- (4 books every 2 hours)
- (2 books sold)

**1. Identify the Average Rate ()**

For our bookstore, the average rate of selling books is 4 every 2 hours.

**2. Determine the Desired Number of Events (k)**

We want to find the probability of selling exactly 2 books, so .

**3. Plug into the Formula**

Remember, the formula is:

**4. Break Down the Formula**

- is the probability of no books being sold.
- is the likelihood of selling the books.
- adjusts for the number of ways 2 books can be sold.

**5. Calculate the Probability**

The result indicates a 14.65% chance that the bookstore will sell exactly 2 books in the next 2 hours.

**Tools for Poisson Distribution Calculation**

When it comes to calculating the Poisson Distribution, there are a plethora of tools available, ranging from software applications to online tools.

**Software**

**Microsoft Excel**: Excel has built-in functions like

that can be used for Poisson calculations.**POISSON.DIST****R**: A popular statistical software where you can use the`dpois`

function for Poisson calculations.**MATLAB**: Offers the`poisspdf`

function for Poisson Distribution.

**Online Tools**

**Stat Trek’s Poisson Calculator**: A simple online tool where you input your lambda value and desired number of events.**Omni Calculator’s Poisson Distribution**: User-friendly interface with clear instructions.**GraphPad’s QuickCalcs**: Offers a Poisson calculator among other statistical tools.

**…So, Which Tool is the Best?**

My recommendation are:

**R**: For those who are serious about statistics and might need to perform a variety of analyses, R is highly recommended. It’s free, open-source, and has a vast community which means lots of resources and tutorials.**Omni Calculator’s Poisson Distribution**: For quick and easy calculations without the need for software installation, this tool is recommended. It’s straightforward and provides clear results.