Exponential distribution
From Maths
Contents
Definition
Let [ilmath]\lambda\in\mathbb{R}_{\ge 0} [/ilmath] be given, and let [ilmath]X\sim\text{Exp}(\lambda)[/ilmath] be an exponentially distributed random variable. Then:
- the probability density function, [ilmath]f:\mathbb{R}_{\ge 0}\rightarrow\mathbb{R}_{\ge 0} [/ilmath] is given as follows:
- [ilmath]f:x\mapsto \lambda e^{-\lambda x} [/ilmath], from this we can obtain:
- the cumulative distribution function, [ilmath]F:\mathbb{R}_{\ge 0}\rightarrow[0,1]\subseteq\mathbb{R} [/ilmath], which is:
- [ilmath]F:x\mapsto 1-e^{-\lambda x} [/ilmath]
- The proof of this is claim 1 below
- [ilmath]F:x\mapsto 1-e^{-\lambda x} [/ilmath]
The exponential distribution has the memoryless property[Note 1]
Notes
- ↑ Furthermore, the memoryless property characterises the exponential distribution, that is a distribution has the memoryless property if and only if it is a member of the exponential distribution family, i.e. an exponential distribution for some [ilmath]\lambda\in\mathbb{R}_{>0} [/ilmath]
