The Poisson distribution is a discrete probability distribution that gives the probability of a given number of events occurring in a fixed interval of time or space, given the events occur with a known constant mean rate and independently of the time since the last event.

If 𝑋 is a Poisson random variable with mean 𝜆, then the probability mass function (PMF) is given by:

𝑃(𝑋=𝑘) = \frac{𝑒^{−𝜆}𝜆^{𝑘}}{𝑘!}, 𝑘 = 0,1,2,…

Where:
• 𝜆 = average number of events in a given interval
• 𝑒 ≈ 2.71828
• 𝑘! = factorial of 𝑘

1. Mean and Variance:
   Mean = 𝜆, Variance = 𝜆
2. Additive Property: If 𝑋₁ ∼ Poisson(𝜆₁) and 𝑋₂ ∼ Poisson(𝜆₂) are independent, then:
   𝑋₁ + 𝑋₂ ∼ Poisson(𝜆₁ + 𝜆₂)
3. Memoryless Nature: Though primarily a property of the exponential distribution, the Poisson process (from which Poisson distribution originates) also assumes independent and memoryless events.
4. Skewness:
   Skewness = \frac{1}{√𝜆}
   This shows that the distribution becomes more symmetric as 𝜆 increases.

The Poisson distribution is derived as a limiting case of the Binomial distribution when:
• Number of trials 𝑛 →∞
• Probability of success 𝑝 → 0
• 𝑛𝑝 = 𝜆 remains constant

Let 𝑋 ∼ Binomial(𝑛,𝑝), then:
𝑃(𝑋 = 𝑘) = \binom{𝑛}{𝑘}𝑝^{𝑘}(1−𝑝)^{𝑛−𝑘}

Taking the limit as 𝑛 → ∞, 𝑝 → 0, such that 𝑛𝑝 = 𝜆, we get:
𝑃(𝑋 = 𝑘) = \frac{𝑛^{𝑘}𝑒^{−𝜆}𝜆^{𝑘}}{𝑘!}.

1. Poisson's Equation in PDEs: The Poisson equation is a partial differential equation of the form:
   ∇²𝜙 = 𝑓(𝑥,𝑦,𝑧)
   It is used in problems involving electrostatics, gravitational fields, and heat conduction, where the source term 𝑓 is often related to a Poisson-distributed phenomenon.
2. Telecommunication: Models the number of phone calls or messages received per unit time.
3. Quality Control: Determines the number of defects in manufactured products.
4. Traffic Flow: Models vehicle arrivals at a traffic intersection.
5. Radiation Physics: Describes the number of radioactive decays in a given time frame.

Feature
Poisson
Binomial
Normal
Type
Discrete
Discrete
Continuous

Domain
𝑘 = 0,1,2,…
𝑘 = 0,1,2,…,𝑛
−∞ < 𝑥 < ∞

Mean
Yes (𝜆)
No
No

Variance
Derived from Binomial (limit case)
Bernoulli trials
Central Limit Theorem

Symmetry
Skewed unless large 𝜆
Approx. symmetric for large 𝑛
Symmetric

Example 1: If the average number of emails received in an hour is 5, what is the probability that exactly 3 emails are received in a particular hour?
Solution: Here, 𝜆 = 5, 𝑘 = 3
𝑃(𝑋 = 3) = \frac{𝑒^{−5}5^{3}}{3!} ≈ 0.1404

Example 2: A manufacturing unit produces a defect on average every 2 meters. Find the probability that there will be no defect in a 4-meter length.
Solution: Rate = 1 defect / 2 meters ⇒ 𝜆 = 4/2 = 2
𝑃(𝑋 = 0) = \frac{𝑒^{−2}2^{0}}{0!} = 𝑒^{−2} ≈ 0.1353