Difference between revisions of "Exponential distribution"
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<strong>Theorem:</strong> Let $X$ have the [[exponential distribution]] on $\mathbb{T}$. Then, | <strong>Theorem:</strong> Let $X$ have the [[exponential distribution]] on $\mathbb{T}$. Then, | ||
| − | $$ | + | $$\mathrm{E}_{\mathbb{T}}(X)=\dfrac{1}{\lambda}.$$ |
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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<strong>Theorem:</strong> Let $X$ have the [[exponential distribution]] on $\mathbb{T}$. Then, | <strong>Theorem:</strong> Let $X$ have the [[exponential distribution]] on $\mathbb{T}$. Then, | ||
| − | $$ | + | $$\mathrm{Var}_{\mathbb{T}}(X)=\dfrac{1}{\lambda^2}.$$ |
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 21:59, 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, $$\mathrm{E}_{\mathbb{T}}(X)=\dfrac{1}{\lambda}.$$
Proof: █
Theorem: Let $X$ have the exponential distribution on $\mathbb{T}$. Then, $$\mathrm{Var}_{\mathbb{T}}(X)=\dfrac{1}{\lambda^2}.$$
Proof: █
References
Probability distributions | ||
| Uniform distribution | Exponential distribution | Gamma distribution |
