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What is the realistic significance of Poisson distribution, and why does real life mostly obey Poisson distribution?

Conclusion first: Because many phenomena in reality are the "accumulation of a large number of independent small probability events".

What is Poisson Distribution?​

Poisson distribution describes the probability distribution of the number of random events occurring within a fixed time or space.

For example:

How many customers came to the coffee shop in this hour?

How many typos are there on this page of the book?

How many accidents happened at this intersection today?

The answers to these questions all follow the Poisson distribution.

Why do so many phenomena follow Poisson distribution?​

Because they all satisfy three conditions:

    1. Independence: The occurrence of each event is independent of each other. Whether this customer comes or not does not affect the next customer.
    1. Stability: The average occurrence rate remains stable during the observation period. It's not like 100 people suddenly come during the morning rush hour, and 1 person comes usually.
    1. Sparsity: At most one event occurs in an extremely short period of time. Two customers will not step into the door at the same second.

Think carefully, many phenomena in life satisfy these three conditions:

  • Customer service calls (each person calling is independent)
  • Website visits (each user click is independent)
  • Traffic accidents (each accident happens independently)
  • Radioactive decay (each atom decay is independent)
  • Typing errors (each error is produced independently)

Intuitive understanding​

Imagine you work in a coffee shop, and on average 3 customers come per hour (λ=3).

Then in this hour:

Probability of 0 people coming ≈ 5%

Probability of 3 people coming ≈ 22% (most likely)

Probability of 10 people coming ≈ 0.08%

Poisson distribution tells us: even if the average is 3, the actual value will fluctuate between 0-7.

This is why sometimes the shop is empty, and sometimes there is a long queue suddenly - this is not luck, it is a mathematical law.

The Essence of Poisson Distribution​

Poisson distribution is actually the limit of binomial distribution.

Divide an hour into 3600 seconds, the probability of a customer entering the door every second is 3/3600 ≈ 0.083%.

When n → ∞, p → 0, but np = λ remains unchanged, the binomial distribution approaches the Poisson distribution.

So the essence of Poisson distribution is: the accumulation of a large number of independent small probability events.

Why learn Poisson Distribution?​

Because it is too useful:

  • Queuing theory: Banks and hospitals use it to predict passenger flow and decide how many windows to open
  • Insurance actuarial: Predict the number of claims and calculate premiums
  • Quality control: Predict the number of product defects
  • Network engineering: Predict server load and do capacity planning
  • Epidemiology: Predict the number of disease cases

Next time you are impatient waiting in line at the bank, you can think: this is Poisson distribution at work.

I made an interactive visualization page where you can adjust the λ value to see the probability distribution change, and verify it with Monte Carlo simulation. You can play with it if you are interested.

Poisson Distribution: Random Laws in Life

Poisson Distribution Visualization Page