Poisson DistributionJEE Advanced

Concept

The Poisson distribution models counts of rare independent events at a fixed average rate λ: calls per hour, misprints per page, decays per second. It is the limit of the binomial with n → ∞, p → 0, np = λ held fixed. Its signature: mean = variance = λ, and everything hangs off the factor e^(−λ).

Key formula

Derivation

From B(n, λ/n): C(n,k)(λ/n)ᵏ(1 − λ/n)^(n−k). As n → ∞, C(n,k)/nᵏ → 1/k! and (1 − λ/n)ⁿ → e^(−λ) — the pmf drops out.

Probabilities sum to 1 because Σλᵏ/k! = e^λ — the exponential series itself. Recursion P(k+1) = P(k)·λ/(k+1) makes the whole distribution computable from P(0) = e^(−λ).

Scenarios to explore

• Poisson Distribution — Rare events at a fixed rate — mean = variance = λ.

Real-world applications

• Radioactive decay counts (physics crossover).
• Network packet arrivals, call-centre traffic.
• Insurance: claims per period; ecology: organisms per plot.

JEE exam tips

• λ scales with the window: 3 calls/hour ⟹ λ = 6 for a two-hour window.
• If a question gives mean = variance, Poisson is being hinted.
• Ratio test P(k+1)/P(k) = λ/(k+1) finds the mode instantly: grows until k < λ.

Common mistakes

• Forgetting e^(−λ) in the pmf.
• Using Poisson when p isn't small or events aren't independent.
• P(X ≥ 1): compute 1 − e^(−λ), not a long sum.

Exam traps to avoid

• Two independent Poissons add: X + Y ~ P(λ₁ + λ₂) — but only if independent.
• Integer λ gives TWO modes: λ and λ − 1.