Poisson Distribution | Data Sc. and A.I. Lect. Series

📘 Understand Poisson Distribution

📌 Introduction

• In binomial distributions, events’ occurrences and non-occurrences are equally important.
• However, in real-life situations:
  • Events do not occur as outcomes of fixed number of trials.
  • Events occur randomly over time.
  • Interest lies only in the number of occurrences.
• Examples:
  • Number of printing mistakes per page in a book.
  • Number of defects in production per item.
  • Number of accidents during a time interval.

🔄 Why Poisson and Not Binomial?

• Poisson is used when:
  • \( n \to \infty \)
  • \( p \to 0 \)
  • \( np = \lambda \), a finite constant
• Introduced by S.D. Poisson in 1837.

📐 Definition of Poisson Distribution

A random variable \( X \) follows Poisson distribution if:

\[
P(X = x) =
\begin{cases}
\dfrac{e^{-\lambda} \lambda^x}{x!}, & x = 0, 1, 2, \dots,\ \lambda > 0 \\
0, & \text{elsewhere}
\end{cases}
\]

• \( \lambda \): mean number of occurrences in a fixed interval.
• \( e \approx 2.7183 \): base of natural logarithm.

📏 Moments of Poisson Distribution

First Moment (Mean)

\[
\mu’_1 = E(X) = \lambda
\]

Second Moment

\[
\mu’_2 = \lambda^2 + \lambda
\]

Variance

\[
\text{Var}(X) = \mu’_2 – (\mu’_1)^2 = \lambda
\]

📊 Higher Moments

Third Moment

\[
\mu’_3 = \lambda^3 + 3\lambda^2 + \lambda
\quad \Rightarrow \quad
\mu_3 = \lambda
\]

Fourth Moment

\[
\mu’_4 = \lambda^4 + 6\lambda^3 + 7\lambda^2 + \lambda
\quad \Rightarrow \quad
\mu_4 = 3\lambda^2 + \lambda
\]

📈 Skewness and Kurtosis

• Skewness:
  \[
  \gamma_1 = \frac{1}{\sqrt{\lambda}}
  \]
• Kurtosis:
  \[
  \gamma_2 = 3 + \frac{1}{\lambda}
  \]
• Poisson distribution is:
  • Positively skewed
  • Leptokurtic when \( \lambda \) is small

📉 Poisson Distribution Table (λ = 3)

x	\(P(X=x)\)
0	0.0498
1	0.1494
2	0.2240
3	0.2240
4	0.1680
5	0.1008
6	0.0504
7	0.0216
8	0.0081
9	0.0027
10	0.0008

📉 Poisson Distribution Table (λ = 10)

x	\(P(X=x)\)
8	0.1121
9	0.1246
10	0.1246
11	0.1132
12	0.0943
13	0.0725
14	0.0518
15	0.0345
16	0.0216

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Poisson Distribution

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