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Erschienen in:
2012 | OriginalPaper | Buchkapitel
2. Poisson Processes
verfasst von : Richard Durrett
Erschienen in: Essentials of Stochastic Processes
Verlag: Springer New York
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Abstract
To prepare for our discussion of the Poisson process, we need to recall the definition and some of the basic properties of the exponential distribution. A random variable T is said to have an exponential distribution with rate λ, or T = exponential(λ), if Here we have described the distribution by giving the distribution function
F(t) = P(T ≤ t). We can also write the definition in terms of the density function
f
T
(t) which is the derivative of the distribution function.
$$\begin{array}{rcl} P(T \leq t) = 1 - {e}^{-\lambda t}\quad \mbox{ for all }t \geq 0& &\end{array}$$
(2.1)
$$\begin{array}{rcl}{ f}_{T}(t) = \left \{\begin{array}{@{}l@{\quad }l@{}} \lambda {e}^{-\lambda t}\quad &\mbox{ for }t \geq 0 \\ 0 \quad &\mbox{ for }t < 0 \end{array} \right.& &\end{array}$$
(2.2)