Stochastic Process

HW4

如有错误，烦请指正.

根据非齐次 Poisson 过程的定义，错误.

由题得

(X (t_{1} + h), X (t_{2} + h), \dots, X (t_{n} + h)) \overset{d}{=} (X (t_{1}), X (t_{2}), \dots, X (t_{n})) ，

此为严平稳过程.

$X$ $\psi(t)$ ，则

\begin{matrix} E (X) = {\frac{d}{d t} ψ (t) |}_{t = 0} = ψ^{'} (0), \\ E (X^{2}) = {\frac{d^{2}}{d t^{2}} ψ (t) |}_{t = 0} = ψ^{″} (0) . \end{matrix}

所以

Var (X) = E (X^{2}) - (E (X))^{2} = ψ^{″} (0) - ψ^{' 2} (0) .

$n$ $[0,t]$ $X_1,X_2,\cdots,X_n$ ，则

\begin{matrix} f_{X_{i}} (x) = f_{X} (x) = \frac{1}{t}, 0 ⩽ x ⩽ t, \\ F_{X} (x) = \int_{0}^{t} f_{X} (x) d x = \frac{x}{t}, 0 ⩽ x ⩽ t . \end{matrix}

$Y=\max\limits_i\set{X_i}$ ，则

F_{Y} (x) = F^{n} (x) = \frac{x^{n}}{t^{n}}, 0 ⩽ x ⩽ t .

根据定理 4，

$[0,t]$ $n$ $n$ $T_n$ $X_{(n)}=\max\limits_i \set {X_i}=Y$ 有着相同分布，即

\begin{matrix} F_{T_{n} | N (t) = n} (x) = F_{Y} (x), \\ \begin{aligned} E (T_{n} | N (t) = n) & = E (Y) \\ = \int_{0}^{t} x d F_{Y} (x) \\ = \int_{0}^{t} x \cdot \frac{n x^{n - 1}}{t^{n}} d x \\ = \frac{n}{t^{n}} \int_{0}^{t} x^{n} d x \\ = \frac{n}{t^{n}} \cdot {\frac{x^{n + 1}}{n + 1} |}_{0}^{t} \\ = \frac{n}{t^{n}} \cdot \frac{t^{n + 1}}{n + 1} \\ = \frac{n t}{n + 1} . \end{aligned} \end{matrix}

以下视“每盏灯烧坏的概率”为强度（速率）.

$N_1(t)$ $N_2(t)$ $\lambda_1=0.02$ $\lambda_2=0.05$ 的 Poisson 过程.

$T_1$ $N_1(t)$ $T_2$ $N_2(t)$ $T_1\sim Exp(0.02)$ $T_2\sim Exp(0.05)$ $E(T_1)=50$ $E(T_2)=20$ .

$T$ $T=T_1+T_2$ $E(T)=E(T_1+T_2)=E(T_1)+E(T_2)=70$ .

所以，

\frac{E (T_{2})}{E (T)} = \frac{20}{70} = \frac{2}{7},

$2/7$ ；

$t=0$ $E(T)=70$ ，即 70 天.