Difference between revisions of "Exponential distribution"
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| − | Let $\mathbb{T}$ be a time scale. Let $\lambda > 0$ and $\ominus \lambda$ be [[positively mu regressive | positively $\mu$-regressive]] and let $t \in \mathbb{T}$. The exponential distribution is given by the [[probability density function]] | + | Let $\mathbb{T}$ be a time scale. Let $\lambda > 0$ and $\ominus \lambda$ be [[positively mu regressive | positively $\mu$-regressive]] constant functions and let $t \in \mathbb{T}$. The exponential distribution is given by the [[probability density function]] |
$$f(t) = \left\{ \begin{array}{ll} | $$f(t) = \left\{ \begin{array}{ll} | ||
-(\ominus \lambda)(t) e_{\ominus \lambda}(t,0) &; t \geq 0 \\ | -(\ominus \lambda)(t) e_{\ominus \lambda}(t,0) &; t \geq 0 \\ | ||
Revision as of 21:58, 14 April 2015
Let $\mathbb{T}$ be a time scale. Let $\lambda > 0$ and $\ominus \lambda$ be positively $\mu$-regressive constant functions and let $t \in \mathbb{T}$. The exponential distribution is given by the probability density function $$f(t) = \left\{ \begin{array}{ll} -(\ominus \lambda)(t) e_{\ominus \lambda}(t,0) &; t \geq 0 \\ 0 &; t<0. \end{array} \right.$$
Properties
Theorem: Let $X$ have the exponential distribution on $\mathbb{T}$. Then, $$\mathbb{E}_{\mathbb{T}}(X)=\dfrac{1}{\lambda}.$$
Proof: █
Theorem: Let $X$ have the exponential distribution on $\mathbb{T}$. Then, $$\mathbb{V}ar_{\mathbb{T}}(X)=\dfrac{1}{\lambda^2}.$$
Proof: █
References
Probability distributions | ||
| Uniform distribution | Exponential distribution | Gamma distribution |
