Poisson Distribution — $ \mathrm{Pois}(\lambda) $
Definition
Parameters:
- Rate $ \lambda > 0 $
Support:
\(x \in \{0,1,2,\dots\}\)
PMF:
\(\mathbb{P}(X=x)
=
\frac{\lambda^x e^{-\lambda}}{x!}\)
Moment Generating Function (MGF) and Characteristic Function
MGF:
\(M_X(t)
=
\exp\!\left(\lambda(e^{t}-1)\right)\)
Characteristic Function:
\(\varphi_X(t)
=
\exp\!\left(\lambda(e^{it}-1)\right)\)
Moments
Mean and Variance
\(\mathbb{E}[X]=\lambda, \quad \operatorname{Var}(X)=\lambda\)
Raw Moments
All moments exist and are obtained via MGF differentiation.
Relationship to Binomial
Poisson Limit Theorem
If $ X_n \sim \mathrm{Bin}(n,p_n) $ with \(n p_n \to \lambda, \quad p_n \to 0,\) then \(X_n \Rightarrow \mathrm{Pois}(\lambda)\)
Exponential Family Representation
\[f(x) = \exp\!\left( x\log\lambda - \lambda - \log x! \right)\]- Natural parameter: \(\eta=\log\lambda\)
- Sufficient statistic: \(T(X)=X\)
Sufficiency and Completeness
For $ X_1,\dots,X_n \stackrel{iid}{\sim} \mathrm{Pois}(\lambda) $:
- Complete and minimal sufficient statistic: \(\sum_{i=1}^n X_i\)
Key for UMVU arguments.
Likelihood and Estimation
Likelihood
\(L(\lambda) = \lambda^{\sum X_i} e^{-n\lambda}\)
Maximum Likelihood Estimator
\(\hat{\lambda} = \frac{1}{n}\sum_{i=1}^n X_i = \bar{X}\)
Fisher Information
\(I(\lambda)=\frac{n}{\lambda}\)
Cramér–Rao Lower Bound (CRLB)
\[\operatorname{Var}(\hat{\lambda}) \ge \frac{\lambda}{n}\]MLE attains the bound.
Bayesian Structure
Conjugate Prior
\(\lambda \sim \mathrm{Gamma}(\alpha,\beta)\)
Posterior
\(\lambda \mid X \sim \mathrm{Gamma} \left( \alpha+\sum X_i,\; \beta+n \right)\)
Closure Properties
Convolution
If $ X \sim \mathrm{Pois}(\lambda_1) $, $ Y \sim \mathrm{Pois}(\lambda_2) $, independent, then \(X+Y \sim \mathrm{Pois}(\lambda_1+\lambda_2)\)
Thinning Property
If $ X \sim \mathrm{Pois}(\lambda) $ and each event is kept with probability $ p $, independently, then \(Y \sim \mathrm{Pois}(p\lambda)\)
Used in:
- Poisson processes
- Queueing theory
- Random graph models
Relationship to Poisson Process
If $ N(t) $ is a Poisson process with rate $ \lambda $:
- $ N(t) \sim \mathrm{Pois}(\lambda t) $
- Interarrival times $ \sim \mathrm{Exp}(\lambda) $
- Arrival times $ \sim \mathrm{Gamma} $
Conditional Structure
If \(X \sim \mathrm{Pois}(\lambda_1), \quad Y \sim \mathrm{Pois}(\lambda_2),\) independent, then \(X \mid (X+Y=n) \sim \mathrm{Bin} \left( n,\frac{\lambda_1}{\lambda_1+\lambda_2} \right)\)
Tail Bounds
Chernoff Bound
For $ X \sim \mathrm{Pois}(\lambda) $, \(\mathbb{P}(X \ge (1+\delta)\lambda) \le \left( \frac{e^\delta}{(1+\delta)^{1+\delta}} \right)^\lambda\)
Key Theorems and Facts (Prelim-Relevant)
- Poisson limit of Binomial
- Closure under addition
- Gamma conjugacy
- Thinning property
- Conditional Binomial structure
Exam Takeaways
- Poisson is the rare-event limit
- Counts over time and space
- Sums add rates
- Conditioning reveals Binomial
- Central to stochastic processes
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