19.X.2. Properties of Poisson Distribution

1. Mean and Variance: Mean = 𝜆, Variance = 𝜆

The Poisson distribution has both its mean (the average number of events that occur) and its variance (the measure of how much the number of events varies) equal to the parameter 𝜆. Therefore, if you have a Poisson distribution where 𝜆 is, say, 5, then both the average number of events occurring and the variability around that average will also be 5. This means that, on average, you'd expect 5 events to occur, and in any given observation period, you can see the number of events fluctuate around that average, following the same 5 value.

Imagine you're at a coffee shop that receives an average of 5 customers every hour. The mean tells you to expect about 5 customers, while the variance indicates that you might see anywhere from 2 to 8 customers in a typical hour. On average, the flow of customers isn't just consistent; it might vary, but typically it'll hover close to that average of 5.

2. Additive Property: If 𝑋 ∼ Poisson(𝜆₁) and 𝑋 ∼ Poisson(𝜆₂) are independent, then: 𝑋₁ + 𝑋₂ ∼ Poisson(𝜆₁ + 𝜆₂)

The additive property of the Poisson distribution states that if you have two independent events, one occurring at a rate of 𝜆₁ and the other at a rate of 𝜆₂, you can combine these two rates into a single Poisson distribution. The resulting distribution will have a mean equal to the sum of the two original means (i.e., 𝜆₁ + 𝜆₂). This is important when analyzing situations with multiple independent sources of events. For instance, if two different machines each produce defects at rates of 2 and 3 per hour, together they will defect at a combined rate of 5 per hour.

Think about two different sources of rainfall in a city, where one area typically gets 3 inches of rain per month, and another area gets 2 inches. If both areas are independently measured, the total rainfall across both areas gives us a new average of 5 inches per month, reflecting the sum of the two averages.

3. Memoryless Nature: Though primarily a property of the exponential distribution, the Poisson process (from which the Poisson distribution originates) also assumes independent and memoryless events.

The memoryless nature implies that the occurrence of an event does not depend on the previous occurrences. This means that the waiting time until the next event is always independent of when the previous event occurred. For example, if you’re waiting for a bus that arrives according to a Poisson process, the time you wait for the next bus does not change regardless of how long you’ve already waited. Each moment is a fresh opportunity for the bus to arrive.

Imagine you're waiting for a shopping transaction to be completed at a busy store. The next customer might come up right after you’ve finished, or you might have to wait a while, but how long you've been waiting already won't affect when the next customer arrives. Each new customer arriving is independent of how long you've been there.

4. Skewness: Skewness = $\frac{1}{\sqrt{\lambda}}$ This shows that the distribution becomes more symmetric as 𝜆 increases.

Skewness is a measure that indicates the asymmetry of a probability distribution. The formula given shows that as 𝜆 increases, the skewness decreases, making the distribution more symmetric. For smaller values of 𝜆, the distribution is positively skewed, meaning it has a longer tail on the right side. As you increase 𝜆, the distribution becomes more balanced and resembles a normal distribution.

Think of a graph showing the number of hits on a website. If your website is newly launched (small λ), you might see many days with zero hits and a few days with hundreds of hits, leading to a skewed distribution. However, as it gains popularity (larger λ), the daily hit counts start to balance out, and the distribution looks more normal and even across days.

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