(1.9) Poisson

Q5.1) How many events can we expect in a fixed interval of time or space? §

Poisson

Let’s say we have a process which we know occurs, on average 5 times per hour. For example, let’s say it is the average number of cars passing a petrol station in the middle of the desert*

The Poisson distribution describes how likely it is, if we ‘observe’ a one hour window, to see no cars, or 1 car, or 2 cars, 3 cars, 4 cars, etc…

Intuitively $5$ (the average) is the most likely number to observe, however it is also very realistic that we could see only 2, 3, 4, or maybe, if we’re very lucky, we could see 10! The Poisson distribution tells us exactly the probability of each of these.

$N\sim Poisson(\lambda =5)$ * What else is there to do in the middle of the desert? ⁑ pronounced ‘Pwah-Son’ ⁂ Means ’fish’ in French

Q5.2) More on Poisson §

At a busy city intersection, on average, 20 cars pass by a coffee shop every minute. The coffee shop owner is considering an advertisement scheme, and estimates that every car has a 40% chance of noticing the shop’s advertisement. Assuming that cars pass by following a Poisson process, and each car independently notices the advertisement with the given probability, what is the expected number of cars noticing the advertisement in 3 minutes?

Explanation

• It just becomes Poisson with a new mean of $20\cdot 0.4$ instead of $20$
• This is known as the ‘thinning’ of a Poisson process

Mathematical Proof (for those interested)

If $N(t)$ represents the number of events up to time $t$, then the distribution of $N(t)$ is Poisson with parameter $\lambda \cdot t$. That is:

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

Q5.3) Even more on Poisson §

Two different websites, Website A and Website B, receive visitors following independent Poisson processes. Website A gets an average of 10 visitors per hour, and Website B gets an average of 15 visitors per hour. If you were to combine the visitor counts from both websites, what would be the expected number of total visitors in 2 hours? Additionally, if the combined visitor counts from both websites follow a Poisson distribution, what is the variance of the total visitors in 2 hours?

Explanation This question is testing the knowledge that the sum of two independent Poisson processes is still a Poisson process. The expected number of visitors in 2 hours for Website A is $2\cdot 10=20$ and for Website B is $2\cdot 15=30$. Thus, the expected combined visitor count is $20+30=50$. Since the sum of two independent Poisson random variables is also Poisson, the variance of the combined count would be the same as its mean, so Variance$=50$.

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The Poisson distribution models the number of events in a fixed interval of time or space. Its PMF is:

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

• $\lambda$ is the average rate of events per interval.
• Expected value (mean) = $\lambda$
• Variance = ( \lambda )

Special Facts

• The sum of Poisson random variables

Q5.2) **Explain how the Poisson distribution arises (hint: Binomial) §

The $Binomial(n,p)$ distribution converges to (becomes) a Poisson distribution as:

• $n\to \infty$
• $p\to 0$
• $np$ (the mean) remains fixed

I.e we allow the number of trials to grow unbounded, while the probability of success of each trial shrinks to 0 proportionally.

Many natural processes have a true underlying Binomial distribution, thus the Poisson distribution is typically used as an approximation to the true underlying distribution.

E.g: when modelling the radioactive decay of a sample of plutonium, you’re really looking at a very large number of individual atoms, each with their own individual and independent extremely small probability of experience radioactive decay in a given time period.

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The Poisson process is a widely used statistical model for counting the number of events occurring within a fixed period of time or space, under certain conditions. It’s particularly useful for modeling rare, independent events in a continuous space. There are several ways to represent and use the Poisson process:

1. Count of Events in a Fixed Interval §

This is the most direct application of a Poisson process. It models the probability of observing a given number of events within a fixed time or space interval. The key parameter is the average rate (λ, lambda) of events per unit of time or space.

• Formula: The probability of observing ( k ) events in a time interval is given by ( P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} ), where ( \lambda ) is the average rate of events per interval, and ( k ) is the number of events.

2. Time Between Events (Exponential Distribution) §

In a Poisson process, the time between successive events follows an Exponential distribution. This aspect models the waiting time until the next event.

• Formula: The probability distribution of the time ( T ) until the next event is ( f(t) = \lambda e^{-\lambda t} ) for ( t \geq 0 ).

3. Cumulative Events Over Time §

This approach models the cumulative number of events up to a certain time. It’s useful for understanding the growth of occurrences over time.

• Representation: A step function that increases by 1 at each event time.

4. Spatial Poisson Process §

The Poisson process can also be applied to spatial data, where events occur in a two-dimensional or three-dimensional space.

• Application: Useful in fields like forestry (modeling the location of trees) or epidemiology (spread of disease in an area).

5. Nonhomogeneous Poisson Process §

In a standard Poisson process, the event rate ( \lambda ) is constant. In a nonhomogeneous process, ( \lambda ) can change over time or space.

• Application: Modeling event rates that vary, such as rush hour traffic or seasonal sales in a store.

6. Poisson Regression §

This is a statistical modeling approach that uses the Poisson distribution to model count data.

• Usage: In scenarios where you want to understand how various factors influence the rate of event occurrence.

7. Queuing Theory §

The Poisson process is often used in queuing theory to model the arrival of customers in a system.

• Application: Service systems like call centers, where customers arrive randomly.

8. Reliability Engineering §

Used to model failure rates in systems over time, assuming the failures follow a Poisson process.

• Example: Predicting the number of system failures in a given time period.

Summary §

The Poisson process is versatile and can be adapted to various fields and scenarios. Its key strength lies in modeling rare, independent, and randomly occurring events over continuous intervals. The selection of the model depends on the specific characteristics of the data and the objectives of the analysis.