Poisson limit of the binomial
Theorem taught in Random variables and distributions, part of Probability
Poisson limit of the binomial
| Type | Theorem |
| Taught in | Random variables and distributions |
| Part | Probability |
| Worked examples | 1 |
| Questions using it | 3 |
If X_n\sim\operatorname{Bin}(n,p_n) with np_n\to\lambda as n\to\infty, then \mathbb{P}(X_n=x)\to e^{-\lambda}\lambda^x/x! for every x=0,1,2,\ldots
Examples
A rare fault
Worked example from Random variables and distributions
A machine makes 1000 parts a day, and each part has a fault with probability 0.002, independently.
Question. Find the probability that a day has at most one faulty part, exactly and by the Poisson approximation.
Solution. The exact count is \operatorname{Bin}(1000,0.002), and \mathbb{P}(X\le1)=0.998^{1000}+1000\times0.002\times0.998^{999}=0.4057. The Poisson approximation uses \lambda=1000\times0.002=2:
\mathbb{P}(X\le1)\approx e^{-2}(1+2)=0.4060.
The two agree to three decimal places. The approximation is good whenever n is large and p is small, and it needs only the product np.
Used in
- Taught in Random variables and distributions, where the derivation and examples are.
- Questions that use it (3) — Particles in an hour (answer) · Groups of three (answer) · C22 (answer)
- All 27 questions on Random variables and distributions
See also
Random variable · Cumulative distribution function · Discrete distribution · Continuous distribution · Expectation · Expectation of a function · Variance · Probability generating function