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Module 9/Normal approximation & the CLT

Poisson approximation & the law of rare events

When successes are rare — large nn, tiny pp — the binomial count is approximately Poisson with mean λ=np\lambda=np. This is the law of rare events, with a clean error bound.

Before you start — give these a try

Attempting first primes your brain for the lesson — even if you miss. Nothing is graded or saved; it's just a warm-up.

X∼Poisson(3)X\sim\mathrm{Poisson}(3). Compute P(X=2)=e−3322!P(X=2)=\dfrac{e^{-3}3^2}{2!}.

Approximating Bin(1000,0.002)\mathrm{Bin}(1000,0.002) by Poisson(2)\mathrm{Poisson}(2), estimate P(X=0)=e−2P(X=0)=e^{-2}.

What you’ll be able to do

  • State the law of rare events: Bin(n,λ/n)→Poisson(λ)\mathrm{Bin}(n,\lambda/n)\to\mathrm{Poisson}(\lambda).
  • Approximate a rare-event binomial by Poisson(np)\mathrm{Poisson}(np) and compute probabilities.
  • Use the error bound ∣P(X∈A)−P(Y∈A)∣≤np2|P(X\in A)-P(Y\in A)|\le np^2.
  • Recognize the regime (large nn, small pp, moderate λ\lambda) where the Poisson approximation applies.

In your course

· MATH2015 · Linear Algebra & Probability
§14.2 Poisson Approximation and the Law of Rare Events
  • Theorem 14.2Law of Rare Events
    lim⁡n→∞P(Sn=k)=e−λλkk!\lim_{n\to\infty}P(S_n=k)=\dfrac{e^{-\lambda}\lambda^k}{k!} for Sn∼Bin(n,λ/n)S_n\sim\mathrm{Bin}(n,\lambda/n).
  • Proposition 14.1Error bound ∣P(X∈A)−P(Y∈A)∣≤np2|P(X\in A)-P(Y\in A)|\le np^2
  • Remark 14.3When to use the Poisson approximation
  • Example 14.2Accidents per month, Poisson(3)\mathrm{Poisson}(3)
  • Example 14.3Typos per page from P(X=0)=0.9P(X=0)=0.9
1

The law of rare events

If successes are very rare in many independent trials, the number of successes is approximately Poisson. Formally, holding the mean λ=np\lambda=np fixed while n→∞n\to\infty (so p=λ/n→0p=\lambda/n\to 0), the Bin(n,p)\mathrm{Bin}(n,p) distribution converges to Poisson(λ)\mathrm{Poisson}(\lambda).

2

Using it in practice

For a binomial with large nn and small pp, approximate X∼Bin(n,p)X\sim\mathrm{Bin}(n,p) by Y∼Poisson(λ)Y\sim\mathrm{Poisson}(\lambda) with λ=np\lambda=np, and read probabilities from\n\nP(Y=k)=e−λλkk!.P(Y=k)=\frac{e^{-\lambda}\lambda^k}{k!}.

3

How good is it? The error bound

Replacing Bin(n,p)\mathrm{Bin}(n,p) by Poisson(np)\mathrm{Poisson}(np) changes the probability of any event by at most np2np^2 (Proposition 14.1). So the approximation is accurate exactly when np2np^2 is small — e.g. nn large and pp very small.

4

When to use it

Rule of thumb (Remark 14.3): n≥100n\ge 100, p≤0.01p\le 0.01, λ=np\lambda=np not too large (typically λ<10\lambda<10), and np2np^2 very small. Classic uses: calls to a call center per hour, accidents under fixed conditions, typos on a page, customers entering a store.

Theorem 14.2 — Law of Rare Events

Let λ>0\lambda>0 and Sn∼Bin(n,λ/n)S_n\sim\mathrm{Bin}(n,\lambda/n). Then for every k∈{0,1,2,… }k\in\{0,1,2,\dots\},\n\nlim⁡n→∞P(Sn=k)=e−λλkk!.\lim_{n\to\infty}P(S_n=k)=\frac{e^{-\lambda}\lambda^k}{k!}.

Intuition. Spreading a fixed expected number λ\lambda of successes over more and more trials, each rarer, gives the Poisson law.
Proposition 14.1 — Error bound

If X∼Bin(n,p)X\sim\mathrm{Bin}(n,p) and Y∼Poisson(np)Y\sim\mathrm{Poisson}(np), then for any A⊆{0,1,2,… }A\subseteq\{0,1,2,\dots\},\n\n∣P(X∈A)−P(Y∈A)∣≤np2.|P(X\in A)-P(Y\in A)|\le np^2.

Intuition. The total error of the Poisson swap is controlled by np2np^2 — small when pp is tiny.

Worked examples

Example 1

A factory averages 33 accidents per month. Modeling the count as Poisson(3)\mathrm{Poisson}(3), find P(X=2)P(X=2).

  1. 1

    With many workers each at small independent risk, Poisson is reasonable; λ=3\lambda=3.

  2. 2

    P(X=2)=e−3322!=9e−32P(X=2)=\dfrac{e^{-3}3^2}{2!}=\dfrac{9e^{-3}}{2}.

  3. 3

    ≈0.224\approx 0.224.

Answer. ≈0.224\approx 0.224.
Example 2

A page has no typos with probability 0.90.9. Estimate P(exactly 2 typos)P(\text{exactly }2\text{ typos}) using a Poisson model.