Exponential distribution/Definition

Grade: B

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use Rice's [ilmath]\mathbb{P} [/ilmath] book. Page 48

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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 on the exponential distribution page

The exponential distribution has the memoryless property