Poisson distribution - A detailed Guide on it

1. What is Poisson distribution?

The Poisson distribution is a probability distribution that simulates the possibility of seeing a certain amount of very rare events takes place over a certain amount of time or space. Its only distinguishing feature is the parameter (lambda), which denotes the average frequency of event occurrences within that interval.

The distribution offers a way to determine the probabilities of observing various numbers of events within the specified interval and is frequently used in circumstances where occurrences are independent and infrequent.

2. Why is it called Poisson distribution?

Siméon Denis Poisson, a French mathematician, is honored by having his name associated with the "Poisson distribution" word.

He developed the distribution to explain the likelihood that rare events will occur during a set period of time, making a contribution to probability theory in the 19th century. Due to his innovative work in this field, the distribution is named in his honor.

3. Why do we use Poisson distribution?

Here is a list of practical applications and reasons for using the Poisson distribution:

A. Counting Events:

The Poisson distribution is frequently used for simulating situations in which events happen at random over a certain amount of time or location.

Examples include the volume of calls a call center receives in an hour, the number of collisions that occur at a certain intersection each day, and the volume of emails that are sent each day.

B. Rare Events:

Modeling rare events that happen rarely but are still important for predicting is made possible by this technique. Equipment malfunctions, unusual diseases, or severe weather may be among those events.

C. Queuing Systems:

The Poisson distribution can be used to estimate the number of customers who will arrive within a specific time frame when evaluating queuing systems, such as the waiting areas at banks or supermarkets.

D. Epidemiology:

In epidemiology, where the number of new cases in a population can be roughly predicted by a Poisson distribution, the distribution is used to model the spread of infectious illnesses.

E. Web Traffic Analysis:

Web traffic analysis can be used to predict how many people will visit, click on, or download content from a website over a given period of time.

F. Radioactive Decay:

Nuclear physics uses the Poisson distribution to describe the quantity of radioactive particles that decay over a certain amount of time.

G. Inventory management:

The distribution helps in managing inventory levels and reorder points in cases where demand for a product is unpredictable and infrequent.

H. Quality Control:

In manufacturing, quality control can be used to examine the occurrence of defects in a production process.

I. Environmental Studies:

The number of species detected in a certain location or the frequency of events like earthquakes or meteorite impacts are both modeled using the Poisson distribution in ecology and environmental science.

J. Financial Risk Analysis:

It can be used to model unpredictable but significant events like spikes in stock prices.

K. Crime Analysis:

To determine the amount of particular crimes in particular locations, law enforcement organizations may use the Poisson distribution.

L. Sports analytics:

In sports, the distribution can be used to model how many goals, points, or scores are scored over the course of a specific amount of time.

M. Service Systems:

Businesses can use the distribution to forecast how many customer contacts or service requests will occur over a specific period of time.

N. Natural Phenomena:

It is used to simulate a variety of natural events, such as meteor showers, lightning strikes, and rainfall events.

O. Accident Analysis:

The distribution is used to study the likelihood that certain accidents, such car accidents or workplace problems, will occur.

4. What is the formula of Poisson distribution?

The probability mass function (PMF) of the Poisson distribution is given by the formula:

P (X = k) = \frac{e^{- λ} . λ^{k}}{k!}

Where:

P(X=k) is the probability of observing k events in a fixed interval of time or space.
e is the base of the natural logarithm, approximately equal to 2.71828.
λ (lambda) is the average rate of occurrence of events in the interval.
k is a non-negative integer representing the number of events you're interested in.

In this formula, e^−λ represents the probability of observing zero events (no events occurring), and $\frac{λ^{k}}{k!}$ represents the probability of observing k events. The combination of these two terms accounts for all possible event counts in the Poisson distribution.

5. What is lambda (λ) in Poisson distribution?

Playing a game where the goal is to catch butterflies in a garden. The term "lambda" (λ) refers to a unique number that indicates how many butterflies are usually caught in a certain period of time. For instance, if λ is 3, it indicates that you typically collect 3 butterflies each hour.

Let's imagine you are interested in finding out your chances of catching a certain quantity of butterflies. In this, the Poisson formula is useful. It's like a magic formula that gives us the chances. It looks like this:

P (C a t h i n g k B u t t e r f l i e s) = \frac{e^{- λ} . λ^{k}}{k!}

Here's what these parts mean:

k is the number of butterflies you want to know about.
λ is the special number that tells us how many butterflies you usually catch.
e is a special number, kind of like 2.71828.
k! means multiplying numbers together from k down to 1.

So, if λ is 3 (you usually catch 3 butterflies in an hour), and you want to know the chance of catching exactly 2 butterflies (k=2), you can use the formula to find out how likely that is.

6. Why is e in Poisson distribution?

Like a special number, "e" allow us to perform mathematical tricks. The hidden element "e" in the Poisson formula is what allows us to accurately predict the frequency of random events like shooting stars and butterflies.

It works like magic math powder to help us in our search for answers!

7. How Poisson distribution is derived?

A. Start with the Binomial Distribution:

The binomial distribution describes the number of successes (k) in a fixed number of trials (n), where each trial has a probability p of success. The probability mass function of the binomial distribution is:

P (X = k) = (\begin{matrix} n \\ k \end{matrix}) . p^{k} . (1 - p)^{n - k}

Here, $(\begin{matrix} n \\ k \end{matrix})$ in the number of ways to choose k successes out of n trials, is the probability of k successes, and $(1 - p)^{n - k}$ is the probability of n-k failures.

B. Take the Limit:

We'll consider what happens when the number of trials (n) becomes very large, and the probability of success (p) becomes very small, while keeping the product np constant. Let's define λ=np, which represents the average number of successes.

C. Simplify Binomial Formula Terms:

As n gets larger, the combination term $(\begin{matrix} n \\ k \end{matrix})$ becomes less important, and the expressions $(1 - p)^{n - k}$ approach . This simplifies the binomial formula:

P (X = k) = (\begin{matrix} n \\ k \end{matrix}) . p^{k} . (1 - p)^{n - k} \approx \frac{n^{k}}{k!} . p^{k} . e^{- λ}

Since λ=np, we have $p - \frac{λ}{n}$ . So, $p^{k} - (\frac{λ}{n})^{k}$ , and $(1 - p)^{n - k} \approx e^{- λ}$

D. Simplify Further:

Now, the term $\frac{n^{k}}{k!}$ behaves similarly to $λ^{k}$ , so we can replace it with $λ^{k}$ in the formula.

This gives us the Poisson distribution formula:

P (X = k) = \frac{e^{- λ} . λ^{k}}{k!}

This derivation shows how, under certain conditions, the Poisson distribution emerges as a simplified version of the binomial distribution when the number of trials becomes very large and the probability of success becomes very small.

8. What are the conditions for a Poisson distribution?

The conditions needed for a distribution to be considered as a Poisson distribution are as follows:

A. Fixed Interval:

The events or occurrences being counted must take place inside a fixed period of time, space, or volume.

B. Independence:

The occurring of one event shouldn't have any effect on the occurrence of another, hence the occurrences should be independent of one another.

C. Rare Events:

Events should be a bit rare or infrequent within the specified time frame. This indicates that there is almost little chance that more than one event will occur in a very small sub-interval.

D. Constant Rate:

Throughout the entire interval, the average rate of events (λ) should be roughly constant. In other words, the rate of occurrence doesn't change over time or space.

E. Countable Events:

The events being counted must be discrete and countable. You can't have fractional or continuous events.

F. No Overlapping Events:

Events should not overlap or coincide. Each event is unique and doesn't affect the occurrence of others within the interval.

G. Uniformity:

The events should be randomly distributed and not follow any specific pattern or trend within the interval.

H. Event Exclusivity:

Each event can only occur once; there is no possibility of multiple occurrences of the same event within the same interval.

8. What is the difference between binomial and Poisson distributions?

Here's a tabular comparison between the binomial and Poisson distributions:

Aspects	Binomial Distribution	Poisson Distributions
Number of Trials	Fixed and finite number of trials (n)	The number of trials (n) becomes very large (n→∞)
Probability of Success	Constant probability (p) of success in each trial	Probability of success (p) is very small
Events Modeled	Multiple successes or failures in a fixed number of trials	Rare events happening in a large number of opportunities
Conditions for Use	Fixed number of trials (n) - Independent events - Constant p	Rare events - Large number of trials (n→∞) - Independent events
Formula Example	$P (X = k) = (\begin{matrix} n \\ k \end{matrix}) . p^{k} . (1 - p)^{n - k}$	$P (X = k) = \frac{e^{- λ} . λ^{k}}{k!}$
Mean and Variance	Mean: np Variance: np(1−p)	Mean: λ Variance: λ
Event Overlapping	Overlapping events are possible	Events are non-overlapping
Distribution Shape	Can be symmetric, skewed, or bimodal depending on p	Skewed to the right
Example Applications	Coin flips, success/failure experiments, surveys, polls	Rare diseases, phone calls, accidents, meteor showers, etc.

9. What are the properties or characteristics of Poisson distribution?

The Poisson distribution's main characteristics are listed below:

A. Discrete Distribution:

The Poisson distribution is a discrete probability distribution, which means it estimates the likelihood that particular discrete events (whole numbers) will take place.

B. Counting Events:

It is used to simulate the number of events that take place over a certain amount of time or space, where each event occurs independently and at random.

C. Parameter:

One parameter, "lambda" (λ), which reflects the average rate of events in the interval, determines the distribution.

D. Rare Events:

The Poisson distribution is particularly helpful for modeling unusual events, because the likelihood of numerous events occurring within a brief period of time is relatively low.

E. Non-Negative Integer Values:

Only non-negative integer values (0, 1, 2......) are possible for the random variable in a Poisson distribution.

F. Mean and Variance:

The mean (μ) and variance $(σ^{2})$ ( ) of the Poisson distribution are both equal to the parameter λ. This is a unique property of the Poisson distribution: $μ = σ^{2} = λ$

G. Skewed Distribution:

Positively skewed distributions have a big tail on the right side, and the Poisson distribution is positively skewed. This is due to the likelihood that incidents will be few in number, even this is not always the case.

H. Probability Mass Function (PMF):

The probability of observing k events in a fixed interval is given by the formula $P (X = k) = \frac{e^{- λ} . λ^{k}}{k!}$ , where e is Euler's number (approximately 2.71828).

I. Independence of Events:

It is considered that events occur independently of one another. One event's occurrence does not influence the occurrence of another event during the same time period.

J. No Memory Property:

The "no memory" property of the Poisson distribution states that the likelihood of witnessing a specific number of events in the future is independent of the events that have already occurred.

K. Binomial distribution approximation:

When the number of trials (n) is very large and the chance of success (p) is very small, the Poisson distribution can be used to approximate the binomial distribution.

10. How to find lambda in Poisson distribution?

The average rate of events happening during a specific period of time or space must be understood in order to determine the value of lambda (λ) in the Poisson distribution. Here is how to calculate the value of λ in a certain situation:

A. Information Provided:

Start by learning as much as you can about the situation you're in. You must be aware of the average rate of the event occurrences throughout the specified time period. This could be events per unit of area (e.g., accidents per junction each day) or events per unit of time (e.g., calls per hour).

B. Count the Events:

The number of events that actually took place during the time period you are interested in is to be counted. Make that the interval used for this count matches the interval used for the Poisson distribution.

C. Calculate the Average:

Subtract the total number of events from the interval's length. You are then provided with the average rate of incidents (λ). $λ = \frac{T o t a l N u m b e r o f E v e n t s}{L e n g t h o f I n t e r v a l}$

D. Use the calculated Lambda:

You can analyze the probabilities of various event counts within that period by using the calculated value of λ as the parameter in the Poisson distribution calculation.

Remember that λ represents the average number of events in the interval. It's important to match the units of λ with the units of the interval you're considering. For example, if you're looking at events per hour, λ should also be expressed as events per hour.

11. When to use Poisson distribution?

Think about counting the number of times a bird flies by in a minute while you're in a park. You might use standard math to calculate the probability of seeing various numbers of birds if the bird flies by frequently and you can count each time it passes.

What if the bird is extremely rare—say, a super exceptional bird that only appears occasionally? The Poisson distribution saves the day at that point! It enables us to understand the probability of observing a specified number of extremely rare events at a particular time or location, such as the exceptional bird flying by.

Hence, the Poisson distribution can be used to determine the probability of various numbers of rare events occurring when something very rare occurs, such as receiving only a few candies while trick-or-treating or observing only a few shooting stars in the sky. It like a magical tool for rare items.

12. How to calculate Poisson distribution?

Let’s go through a simple example step by step:

Imagine you're counting shooting stars in the sky during a meteor shower. On average, you see about 5 shooting stars per hour. Now, you want to know the chances of seeing exactly 3 shooting stars in the next hour.

Here's how you can use the Poisson distribution to figure it out:

A. Average Shooting Stars (λ):

You know you usually see 5 shooting stars in an hour. So, your "lambda" (λ) is 5.

B. Number You Want to Know (k):

You want to know the chances of seeing exactly 3 shooting stars.

C. Plug into the Formula:

Use the Poisson formula:

Chances of seeing 3 shooting stars = $\frac{e^{- 5} . 5^{3}}{3!}$

D. Calculate:

C h a n e s i f s e e i n g 3 S h o o t i n g s t a r s = \frac{0.067 * 125}{6} = 0.139

So, the chances of seeing exactly 3 shooting stars in the next hour during the meteor shower are about 0.139, which is roughly 13.9%.

13. What is the mean of Poisson distribution?

The mean of a Poisson distribution is the average number of events expected to occur in a fixed interval. It's represented by the parameter "lambda" (λ) and is also equal to the variance of the distribution.

14. How to do Poisson distribution?

To work with the Poisson distribution, follow these steps:

A. Identify λ (Average Rate):

Determine the average rate of events (λ) occurring in the given interval (time, space, etc.).

B. Define the Event:

Decide what specific event you're interested in counting or observing.

C. Use the Poisson Formula:

Apply the Poisson distribution formula:

P (X = k) = \frac{e^{- λ} . λ^{k}}{k!}

Where k is the number of events you want to find the probability for.

D. Plug in Values:

Substitute λ (average rate) and k (the specific number of events) into the formula.

E. Calculate the Probability:

Do the calculations using the formula, e (around 2.71828), and factorials.

F. Interpret the Result:

The calculated probability (P(X=k)) represents the chances of observing k events in the given interval.

Remember that λ is key; it tells you the average rate of events happening. The Poisson distribution helps you understand the likelihood of observing different numbers of events based on that average rate.

Poisson distribution - A detailed Guide on Poisson distribution