The [[Poisson Distribution]] arises as a limiting case of the [[Binomial Distribution]]. Consider a [[Random Variable]] $S \sim \mathrm{Binom}(n,p)$, where: - $n \to \infty:$ Number of trials becomes very large. - $p\to 0:$ Probability of success in each trial becomes very small. - $np=\lambda:$ Expected number of successes. $\lambda$ needs to remain moderate. In this scenario the approximation holds and is expressed as: $ \mathrm{Binom}(n,p) \xrightarrow[n \to \infty, \:p \to 0]{(d.)} \mathrm{Poisson}(\lambda)$ The [[Probability Mass Function|PMF]] of the Binomial distribution converges to the PMF of the Poisson distribution: $ p_S(k) = \frac{\lambda^k}{k!}*e^{-\lambda}$ ## Derivation The derivation starts with the PMF of the Binomial random variable $S$. $ \begin{align} p_S(k) &=\frac{n!}{(n-k)!*k!}*p^k * \Big(1-p\Big)^{n-k} \tag{1}\\[10pt] &=\frac{n!}{(n-k)!*k!}*\frac{\lambda^k}{n^k} * \left(1-\frac{\lambda}{n}\right)^{n-k} \tag{2}\\[10pt] &=\frac{n*(n-1)* \dots * (n-k+1)}{k!}*\frac{\lambda^k}{n^k} * \left(1-\frac{\lambda}{n}\right)^{n-k} \tag{3}\\[10pt] &=\frac{n_(n-1)* \dots * (n-k+1)}{n^k}*\frac{\lambda^k}{k!} * \left(1-\frac{\lambda}{n}\right)^{n-k} \tag{4}\\[10pt] &=\frac{n}{n}*\frac{(n-1)}{n}* \dots * \frac{(n-k+1)}{n}*\frac{\lambda^k}{k!} * \left(1-\frac{\lambda}{n}\right)^{n} * \left(1-\frac{\lambda}{n}\right)^{-k} \tag{5}\\[10pt] \lim_{n\to \infty} &= 1*1* \dots * 1* \frac{\lambda^k}{k!}*e^{-\lambda}*1 \tag{6}\\[10pt] p_S(k) &\approx \frac{\lambda^k}{k!}*e^{-\lambda} \tag{7} \end{align} $ where: - (1) [[Probability Mass Function|PMF]] of the Binomial r.v. $S$. - (2) Exchange $p$ with $\lambda \over n$. - (3) Dividing the two factorial series $n!$ and $(n-k)!$ results in a shortened factorial series. - (4) Switch denominators $k!$ with $n^k$. - (5) Split up the exponent of $(1-\frac{\lambda}{n})$. - (6) Evaluate the equation when $n\to \infty$ and $p \to 0$. We utilize the below fact under these circumstances.. $ \lim_{n \to \infty}\Big(1- \frac{\lambda}{n}\Big)^n = e^{-\lambda} $ ## Discretization and Poisson Approximation To better understand the approximation, consider discretizing a time interval $\tau$ into $n$ slots, each with a length of $\delta$. ![[poisson-distribution.png|center|400]] $ \delta=\frac{\tau}{n}, \quad n=\frac{\tau}{\delta}, \quad p=\lambda \delta$ Each slot is treated as a [[Bernoulli Distribution|Bernoulli]] trial, with probability of success $p=\lambda \delta$. This is because probability of success over the full interval $\tau$ is $\lambda$ and we assumed constant success rate over time. As $n \to \infty$, we effectively shorten each slot $\delta \to 0$, which ensures that: - The probability of gt;1$ success in a slit is negligible. - The number of successes over interval $\tau$ is approximated by $\lambda$. **Expectation:** $ \mathbb E[S] =np=\frac{\tau}{\delta} * \lambda\delta=\lambda \tau$ **Variance:** $ \begin{align} \mathrm{Var}(\mathrm{Binom})&= np(1-p) \to np=\lambda, && \text{where} \: p\to 0 \\ \mathrm{Var}(\mathrm{Pois})&=\lambda, && \text{where} \: \lambda = np \end{align} $ ## Comparisons of Approximations The table below summarizes how different approximations arise depending on the parameters of the Binomial distribution. | $n$ | $p$ | $\lambda$ | Approximation | Comment | | ------------ | -------- | ------------ | ------------------ | ------------------------------------------------------------------------------------------------ | | $\to \infty$ | $\to 0$ | Moderate | Poisson | Distribution is too asymmetric for a [[Gaussian Distribution\|Gaussian]] approximation. | | $\to \infty$ | $\to 1$ | $\to \infty$ | Gaussian | Variance is too large for Poisson. | | $\to \infty$ | Moderate | Large | Poisson & Gaussian | Small $p$ captures Poisson variance and large $n$ balances asymmetry for Gaussian approximation. |