3.4.7 Poisson random Process

\(N\left ( t\right ) \) is a Poisson random variable if

1. \(N\left ( 0\right ) =0\)
2. Independent increments
3. \(P\left ( N\left ( t\right ) =n\right ) =\frac {\left ( \lambda t\right ) ^{n}e^{-\left ( \lambda t\right ) }}{n!}\)

Where \(\lambda \) is the average number of events that occur in one unit time. So \(N\left ( t\right ) \) is random variable which is the number of events that occur during interval of length \(t\)

The probability that ONE event occure in the next \(h\) interval, when the interval is very small, is \(\lambda h+o\left ( h\right ) \)

This can be seen by setting \(n=1\) in the definition and using series expansion for \(e^{-\left ( \lambda h\right ) }\) and then letting \(h\rightarrow 0\)

Expected value of Poisson random variable: \(E\left ( N\right ) =\lambda \). For a process, \(E\left ( N\left ( t\right ) \right ) =\lambda t\) where \(\lambda \) is the rate.