Binomial Distribution Approximated by Poisson Distribution
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Theorem
Let $X$ be a discrete random variable which has the binomial distribution with parameters $n$ and $p$.
Then for $\lambda = n p$, $X$ can be approximated by a Poisson distribution with parameter $\lambda$:
- $\ds \lim_{n \mathop \to \infty} \binom n k p^k \paren {1 - p}^{n - k} = \frac {\lambda^k} {k!} e^{-\lambda}$
Proof
Let $X$ be as described.
Let $k \in \Z_{\ge 0}$ be fixed.
We write $p = \dfrac \lambda n$ and suppose that $n$ is large.
Then:
\(\ds \lim_{n \mathop \to \infty} \binom n k p^k \paren {1 - p}^{n - k}\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \binom n k \paren {\frac \lambda n}^k \paren {1 - \frac \lambda n}^n \paren {1 - \frac \lambda n}^{-k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \frac {n^k} {k!} \paren {\frac \lambda n}^k \paren {1 - \frac \lambda n}^n \paren {1 - \frac \lambda n}^{-k}\) | Limit to Infinity of Binomial Coefficient over Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \frac 1 {k!} \lambda^k \paren {1 + \frac {-\lambda} n}^n \paren {1 - \frac \lambda n}^{-k}\) | canceling $n^k$ from numerator and denominator | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\lambda^k} {k!} \lim_{n \mathop \to \infty} \paren {1 + \frac {-\lambda} n}^n \paren {1 - \frac \lambda n}^{-k}\) | moving $\dfrac {\lambda^k} {k!}$ outside of the limit | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\lambda^k} {k!} \lim_{n \mathop \to \infty} \paren {1 + \frac {-\lambda} n}^n \lim_{n \mathop \to \infty} \paren {1 - \frac \lambda n}^{-k}\) | Combination Theorem for Limits of Functions/Product Rule | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\lambda^k} {k!} \lim_{n \mathop \to \infty} \paren {1 + \frac {-\lambda} n}^n \lim_{n \mathop \to \infty} \paren {1 - \frac k n}^{-k}\) | $\lambda = n p$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\lambda^k} {k!} e^{-\lambda} \paren {1 }\) | Definition of Exponential Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\lambda^k} {k!} e^{-\lambda}\) |
Hence the result.
$\blacksquare$
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 2.2$: Examples
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): binomial distribution
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): binomial distribution