Why Poisson Distribution is Limiting Case of Binomial

In this class, We discuss Why Poisson Distribution is Limiting Case of Binomial.

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The reader should have prior knowledge of binomial to Poisson distribution proof. Click Here.

In the previous class, proof of binomial to Poisson distribution, our assumption n -> ∞ , and p –> 0.

Why do we make assumptions? We will understand in this class.

We take the binomial distribution data analytics example.

We posted a video on youtube, and we are getting an average of two likes per day.

λ = two likes per day.

If the question P(X = 4), i.e., the probability of getting four likes per day?

Assume we have an only binomial distribution. Poisson distribution is not invented.

Now we need to solve the binomial distribution.

We need n and p values.

We had the λ value. So we assume some n and p values.

Assume one view in one hour.

Our assumption n = 24 views per day.

Important:

In binomial distribution, we said our experiments are independent.

Getting a view is our experiment.

To make our experiments independent, we assume one view in one hour is wrong.

We get multiple views in one hour.

Assume one view in one minute.

Still, there is a possibility of getting multiple views in one minute.

Assume one view in one second.

Still, there is a possibility of getting multiple views in one second.

Keep dividing time to make views independent of each other.

One day we keep on dividing the time.

n events happen in one day, so n -> ∞.

As n -> ∞ we take p -> 0 why?

We had the value λ

λ = 2 likes per day. 2 a finite value

λ = np

As n -> ∞ , the p-value should be near zero to make np a finite value.