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Poisson Distribution, Poisson Processes

约 125 个字 预计阅读时间 1 分钟

定义

Poissen Distribtion:

对 \(k \in \mathbb{Z}\)，定义泊松分布 \(X \sim \mathrm{Pois}(\lambda)\)

\[ \mathrm{Pr}[X=k] = \begin{cases} e^{-\lambda}\frac{\lambda^k}{k!}, &k \ge 0 \\ 0, & k < 0 \end{cases} \]

容易得到，

\[ E[X] = Var[X] = \lambda \]

Poissen Process:

定义 Poissen Process \(\{N(s): s \ge 0\}\) with rate \(\lambda\)：

  1. \(N(0) = 0\)
  2. \(\forall t,s \ge 0, N(t+s) - N(s) \sim \mathrm{Pois}(\lambda t)\)
  3. \(\forall t_0 \le t_1 \le \cdots \le t_n,\space N(t_1) - N(t_0), N(t_2) - N(t_1), \ldots, N(t_n) - N(t_{n-1})\) 互相独立

Exponential distribution with rate \(\lambda\):

\[ f(x) = \begin{cases} \lambda e^{-\lambda x} &x \ge 0\\ 0 & x < 0\\ \end{cases} \]
  • CDF: \(F(x) = 1 - e^{-\lambda x}\)
  • \(E[X] = \frac{1}{\lambda}, Var[X] = \frac{1}{\lambda ^2}\)

命题

命题1

对 \(X_1 \sim \mathrm{Pois}(\lambda_1), \space X_2 \sim \mathrm{Pois}(\lambda_2)\)，有 \(X_1 + X_2 \sim \mathrm{Pois}(\lambda_1 + \lambda_2)\)

命题2

Suppose that \(\tau_1, \tau_2, \ldots, \tau_n\) is a sequence of independent random variable that \(\tau _i \sim \mathrm{Exp}(\lambda)\). Let \(T_n = \sum_{i=1}^{n}\tau _i\). For \(s \ge 0\), let \(N(s) = max\{ n\mid T_n \le s\}\), then \(N(s)\) is a Poisson process with rate \(\lambda\)

命题3

对 \(X \sim \mathrm{Exp}(\lambda)\)，则对任意 \(t,s \ge 0\)

\[ \mathrm{Pr}[X > t + s | X > s] = \mathrm{Pr}[X > t] \]

命题4

对 \(X_1 \sim \mathrm{Exp}(\lambda_1), \space X_2 \sim \mathrm{Exp}(\lambda_2)\)，有 \(\min(X_1, X_2) \sim \mathrm{Exp}(\lambda_1 + \lambda_2)\)

Thinning

对一段时间内“到达”的泊松随机过程，可以做分割（举例：排队、零件分级等），分为若干子泊松随机过程，则这些过程相互独立。