Poisson Process

Description:

• Counting Process for Poisson distribution
  • Counting how many poisson event has happen before
• The counting process $\{N(t),t\ge 0\}$ is said to be a Poisson process with rate $\lambda >0$ if the following axioms holds:
  1. $N(0)=0$
  2. $\{N(t),t\ge 0\}$ has independent increments
  3. $P(N(t+h)-N(t)=1)=\lambda h+o(h)$
     • If $X$ is a continuous variable with density $f$ and failure rate function $\lambda (t)$
     • then $P(t<X<t+h)\approx f(t)h=f(t)h+o(h)$
       • $f(t)$ represents the probability of failture happening at $t$, $o(h)$ is then probability of failure happening more than 1
     • $P(t<X<t+h|X>t)=\lambda (t)h+o(h)$
  4. $P(N(t+h)-N(t)\ge 2)=o(h)$
• Define $N_s(t)$ for $t>0$, $N_s(t)=N(s+t)-N(s)$ then $\{N_s(t),t\ge 0\}$ is a poisson process with rate $\lambda$
  • nb of events occur in interval $t$
• Let $T_1$ denote the time of the first event of Poisson process $\{N(t),t\ge 0\}$, $T_1=\min \{t\ge 0,N(t)=1\}$
  • $P(T_1>t)=P[N(t)=0]=e^{-\lambda t}$
  • Also defined $T_n$ as the time between the $n-1$th and the $n$ event.
• The sequence $\{T_n,n=1,2,...\}$ is the sequence of interarrival times
  • $T_1,T_2,..$ are independent and identically distributed Exponential random variable with rate $\lambda$
  • They are identical variables
• Let $S_n$ be the time arrival of $n$-th event, then $S_n=\sum _{i=1}^n{T}_i$
  • $S_n$ is then Gamma distribution with PDF: $f_{S_n}(s)=\lambda e^{-\lambda s}\frac{(\lambda s{)}^{n-1}}{(n-1)!}$
    • amount of time for $n$ events to occurs knowing that there is $\lambda$ events occurs
• theorem If $\{N(t),t\ge 0\}$ is a Poison process with rate $\lambda$ then $N(t)$ is a Poisson random variable with rate $\lambda t$
  • That is, $P(N(t)=n)=\frac{e^{-\lambda t}(\lambda t{)}^n}{n!}$ for $n\ge 0$
  • Remarks:
    • The number of events in any fixed interval of length $t$ is Poisson with rate $\lambda$
    • Poisson process has Stationary increments

Further properties:

• Consider a poisson process $\{N(t),t\ge 0\}$ having rate $\lambda$
• suppose that each time an event occurs it is classified as either a type 1, with probability $p$, or a type 2 event, with probability $1-p$, independently of all other events.
• Let $N_1(t)$ and $N_2(t)$ denote respectively the number of type 1 and type 2 events occurring in [0, t], $N(t)=N_1(t)+N_2(t)$
• $\{N_1(t),t\ge 0\}$ and $\{N_2(t),t\ge 0\}$ are both Poisson processes having respective rates $\lambda p$ and $\lambda (1-p)$
• Furthermore, the two processes are independent

Conditional Distribution of the Arrival Times:

• Suppose we know that exactly one event of a Poisson process has taken place by time $t$
• We want to determine the distribution of the time at which the event occurred. We can check, for range $s\le t$, $P\{T_1<s|N(t)=1\}=\frac{s}{t}$
  • This means, the time of the event should be uniformly distributed over [0, t]
• Let $Y_1,Y_2,...,Y_n$ be n random variables.
  • We say that $Y_{(1)},Y_{(2)},...,Y_{(n)}$ are the order statistics corresponding to $Y_1,Y_2,...,Y_n$ if $Y_{(k)}$ is the $k$th smallest value among $Y_1,...,Y_n$, $k=1,2,...,n$
• If the $Y_i,i=1,...,n$, are independent identically distributed continuous random variables with probability density $f$, then the joint density of tthhe order statistics $Y_{(1)},Y_{(2)},...,Y_{(n)}$ is given by $f(y_1,y_2,...,y_n)=n!{\prod }_{i=1}^{n}f(y_i),y_1<y_2<\cdot \cdot \cdot <y_n$
• theorem Given that $N(t)=n$, the $n$ arrival times $S_1,...,S_n$ have the same distribution as the order statistics corresponding to $n$ independent random variables uniformly distributed on the interval $(0,t)$
• Remarks:
  • The preceding result is usually paraphrased as stating that, under the condition that $n$ events have occurred in $(0,t)$, the times $S_1,...,S_n$ at which events occur, considered as unordered random variables, are distributed independently and uniformly in the interval $(0,t)$
• Proposition: If $N_i(t),i=1,...,k$, represents the number of type $i$ events occurring by time $t$ then $N_i(t),i=1,...,k$, are independent Poisson random variables having means $E[N_i(t)]=\lambda {\int }_0^t{P}_i(s)ds$