Difference between revisions of "Exponential distribution"

From timescalewiki
Jump to: navigation, search
(Properties)
 
(4 intermediate revisions by the same user not shown)
Line 1: Line 1:
 +
__NOTOC__
 
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]]
 
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}
Line 6: Line 7:
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
+
[[Expected value of exponential distribution]]<br />
<strong>Theorem:</strong> Let $X$ have the [[exponential distribution]] on $\mathbb{T}$. Then,
+
[[Variance of exponential distribution]]
$$E_{\mathbb{T}}(X)=\dfrac{1}{\lambda}.$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> Let $X$ have the [[exponential distribution]] on $\mathbb{T}$. Then,
 
$$Var_{\mathbb{T}}(X)=\dfrac{1}{\lambda^2}.$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
 
=References=
 
=References=
Line 26: Line 14:
  
 
{{:Probability distributions footer}}
 
{{:Probability distributions footer}}
 +
 +
[[Category:Definition]]

Latest revision as of 14:07, 28 January 2023

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

Expected value of exponential distribution
Variance of exponential distribution

References

[1]

Probability distributions

Uniform distributionExponential distributionGamma distribution