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ไน ้ข˜ไธ‰

้šๆœบ่ฟ‡็จ‹(่‹ไธญๆ น) ็ฌฌไธ‰็ซ  Poisson ่ฟ‡็จ‹ไน ้ข˜

1.

  • \[ET(t)=ET_0E(-1)^{N(t)}=0\]
  • \[Cov(T(s),T(t))=ET_0^2E(-1)^{N(s)}(-1)^{N(t)}=E(-1)^{2N(s)} E(-1)^{N(t-s)}=e^{-2\lambda (t-s)}\]

  • ๅŒ็†ๆ˜“็Ÿฅ

  • \[EX(t)=0\]

  • ่ฎพ \(s<t\)

\[ \begin{align*} Cov(X(s),X(t)) &= ET_0^2\int_0^t\int_0^sE(-1)^{2N(\min \{u,v\})}E(-1)^{N(\max \{u,v\}-\min \{u,v\})}dudv\\ &= \int_0^s du\int_0^u e^{-2\lambda (u-v)}dv + \int_0^sdu\int_u^t e^{-2\lambda (v-u)}dv\\ &=\frac{-1+e^{-2\lambda s} + e^{-2\lambda t}+ 4\lambda s- e^{-2\lambda (t-s)}}{4\lambda^2} \end{align*} \]

2.

  • \[N(t)\sim \mathcal{P}(4t)\]
  • \[p(N(0.5)=1)=\frac{4\cdot 0.5}{1!}e^{-4\cdot 0.5}=2e^{-2}\]
  • \[p(N(2.5)=5)=\frac{(4\cdot 2.5)^5}{5!}e^{-4\cdot 2.5}=\frac{2500}{3}e^{-10}\]

3.

  • \[N(t)\sim \mathcal{P}(2t)\]
  • \[p(N(1)=2)=\frac{2^2}{2!}e^{-2}=2e^{-2}\]
  • \[p(N(1)=2,N(3)=6)=p(N(1)=2,N(3)-N(1)=4)=p(N(1)=2)p(N(2)=4)=2e^{-2}\frac{(4)^4}{4!}e^{-4}=\frac{64}{3}e^{-6}\]
  • \[p(N(1)=2|N(3)=6)=\frac{p(N(1)=2,N(3)=6)}{p(N(3)=6)}=\frac{\frac{64}{3}e^{-6}}{\frac{(2\cdot 3)^6}{6!}e^{-6}}=\frac{80}{243}\]
  • \[p(N(3)=6|N(1)=2)=p(N(2)=4)=\frac{32}{3e^4}\]

4.

  • \[N(t)\sim \mathcal{P}(2t)\]
  • \[p(N(1)\leq 2)=p(N(1)=0)+p(N(1)=1)+p(N(1)=2)=e^{-2}+2e^{-2}+2e^{-2}=5e^{-2}\]
  • \[p(N(1)=1,N(2)=3)=p(N(1)=1)p(N(1)=2)=3e^{-2}\]
  • \[p(N(1)\geq 2|N(1)\geq 1)=\frac{p(N(1)\geq 2,N(1)\geq 1)}{p(N(1)\geq 1)}=\frac{p(N(1)\geq 2)}{p(N(1)\geq 1)}=\frac{1-p(N(1)=1)-p(N(1)=0)}{1-p(N(1)=0)}=\frac{e^2-3}{e^2-1}\]

5.

  • \[N(t)\sim \mathcal{P}(2t)\]
  • \[EN(2)=4\]
  • \[E(N(1))^2=Var(N(1))+(EN(1))^2=2+2^2=6\]
  • \[E(N(1)N(2))=E(N(1)(N(2)-N(1)+N(1)))=(EN(1))^2+E(N(1)^2)=4+6=10\]
  • \[E(N(1)N(2)N(3))=E(N(1)N(2)(N(3)-N(2)+N(2)))=EN(1)N(2)EN(1)+EN(1)N(2)^2=EN(1)N(2)EN(1)+( EN(1)[N(1)+(N(2)-N(1))]^2 )=EN(1)N(2)EN(1)+EN(1)^3+2EN(1)^2EN(1)+EN(1)EN(1)^2 =78\]
  • ๅ…ถไธญ\(\(EN(1)^3=\lambda^3+3\lambda^2+2\lambda|_{\lambda =2}=22\)\)
  • ่ฟ™้‡Œไธ่ƒฝๅฏน\(N(1)\)ๅšๅ˜ๆข,ๅ› ไธบ\(N(2),N(2)-N(1)\)ไธ็‹ฌ็ซ‹,ไฝ†ๆ˜ฏ\(N(1),N(2)-N(1)\)ๆ˜ฏ็‹ฌ็ซ‹็š„

6.

  • \[N(t)\sim \mathcal{P}(2t),T\sim U(1,2)\]
  • \[E(N(T)|T=t)=E(N())=2t,E(N(T)^2|T=t)=4t^2+2t\]
  • \[E(N(T))=\int_1^2\frac{1}{2-1}2tdt=3,Var(N(T))=\int_1^24t^2+2tdt=\frac{31}{3}\]

7.

  • \[N(t)\sim \mathcal{P}(\lambda t)\]
  • \[p(T=t)=p(N(t)\leq k)=\sum\limits_{i=0}^k\frac{(\lambda t)^i}{i!}e^{-\lambda t}\]

8.

  • \[N(t)\sim \mathcal{P}(2t)\]
  • \[p(N(3)=0,N(5)\geq 1)=p(N(3)=0)p(N(5)-N(3)\geq 1)=p(N(3)=0)(1-p(N(2)=0))=e^{-6}(1-e^{-4})\]
  • \[p(N(3)=0,N(5)=1)=p(N(3)=0)p(N(2)=1)=e^{-6}\frac{4}{1}e^{-4}=4e^{-10}\]

9.

  • \[N_\alpha(t)\sim\mathcal{P}(\lambda_1 t),N_\beta(t)\sim\mathcal{P}(\lambda_2 t),N_\gamma(t)\sim\mathcal{P}(\lambda_3 t)\]
  • \[p(T=t)=p(N_\alpha(t)\geq 1,N_\beta(t)\geq 1,N_\gamma(t)\geq 1)=(1-p(N_\alpha(t)=0))(1-p(N_\beta(t)=0))(1-p(N_\gamma(t)=0))=(1-e^{-\lambda_1 t})(1-e^{-\lambda_2 t})(1-e^{-\lambda_3 t})\]

10.

  • ๆ•™ๆ p56 ้กตๆณจ่ฎฐ3.9
  • ็›ดๆŽฅ็”จๅฏ†ๅบฆๅ‡ฝๆ•ฐ็งฏๅˆ†ๆˆ–่€…็”จ็ป“่ฎบ:\(\int_0^t s\times \frac{1}{t^n}\frac{n!}{(k-1)!(n-k)!}s^{k-1}(t-s)^{n-k}|_{n=k=3}ds=\frac{k}{n+1}t|_{n=k=3}=\frac{3}{4}t\)

11.

  • \[E(S_1S_2|N(1)=2)=E(U_{(1)}U_{(2)})=EU_1U_2=\int_0^1\int_0^1xydxdy=\frac{1}{4}\]
  • \[E(S_1+\cdots+S_5|N(1)=5)=E(\sum\limits_{i=1}^5U_{(i)})=E(\sum\limits_{i=1}^5U_i)=\sum\limits_{i=1}^5EU_i=\frac{5}{2}\]

12.

  • \(p(x,y,z)=3!,0<x<y<z<1\),ๅ…ถไธญ\(x,y,z\)่กจ็คบ\(S_1,S_2,S_3\)็š„ๅˆฐ่พพๆ—ถ้—ด,ๅšๅ˜ๆข\(u=x/y,v=(1-x)/(1-y),w=1-y\)ๅˆ™\(|\frac{\partial{(x,y,z)}}{\partial(u,v,w)}|=w-w^2\),ไปŽ่€Œ\(p_{U,V,W}(u,v,w)=3!(w^2-w),p_{U,V}(u,v)=\int_0^13!(w-w^2)dw=1\)

13.

  • \[p(W>w)=p(\min\limits_{n\geq 1}\{S_n+Z_n\}> w)=p(S_1+Z_1>w,\cdots,S_n+Z_n>w)=p(U_1+Z_1>w,\cdots,U_n+Z_n>w)=[1-p(U_1+Z_1\leq w)]^n=[1-p(U_1\leq w-Z_1)]^n=[1-\int_0^wp_{S_1}(w-u)p(u)du]^n=[1-\frac{1}{t}\int_{0}^{w}p(u)du]^n\]

14.

  • \[E(S_1|N(t)=2)=\frac{1}{3}t\]
  • \[E(S_3|N(t)=5)=\frac{1}{2}t\]

15.

  • \[E(S_l-S_k|N(t)=n)=E(U_{(l)}-U{(k)})=EU_{(l)}-EU_{(k)}=\frac{l}{n+1}t-\frac{k}{n+1}t=\frac{l-k}{n+1}t\]
  • \[E(t-S_k|N(t)=n)=t-E(S_k|N(t)=n)=t-\frac{k}{n+1}t=\frac{n-k+1}{n+1}t\]

16.

  • \[EZ(t)=E\sum\limits_{k=1}^{N(t)}\xi_ke^{-\gamma(t-S_k)}=\sum\limits_{n=0}^{\infty}E(\sum\limits_{k=1}^n\xi_ke^{-\gamma(t-S_k)})p(N(t)=n)=\sum\limits_{n=0}^{\infty}p(N(t)=n)\sum\limits_{k=1}^nE\xi_kEe^{-\gamma(t-S_k)}$$,ๅ…ถไธญ$$\sum\limits_{k=1}^nE\xi_kEe^{-\gamma(t-S_k)}=\mu E\sum\limits_{k=1}^ne^{-\gamma(t-U_k)}=\frac{\mu n}{\gamma t}(1-e^{-\gamma t})$$,ๆ•…$$EZ(t)=\frac{\mu}{\gamma t}(1-e^{-\gamma t}) \sum\limits_{n=0}^{\infty}np(N(t)=n) = \frac{\lambda\mu}{\gamma}(1-e^{-\gamma t})\]
  • \[EZ(t)^2=E(\sum\limits_{k=1}^{N(t)}\xi_ke^{-\gamma(t-S_k)})^2=\sum\limits_{n=0}^{\infty}E(\sum\limits_{k=1}^n\xi_ke^{-\gamma(t-S_k)})^2p(N(t)=n)= \sum\limits_{n=0}^\infty \{ E\sum\limits_{k=1}^n\xi_k^2e^{-2\gamma(t-S_k)} +E\sum\limits_{i\neq j}\xi_i\xi_je^{-\gamma(2t-S_i-S_j)} \} = \frac{\lambda(\sigma^2+\mu^2)}{2\gamma}(1-e^{-2\gamma t})+\frac{\mu^2\lambda^2}{\gamma^2}(1-e^{-\gamma t})^2$$,ๆ•… $$VarZ(t)=EZ(t)^2-(EZ(t))^2=\frac{\lambda(\sigma^2+\mu^2)}{2\gamma}(1-e^{-2\gamma t})\]