Difference between revisions of "Expected value"
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$$\mathbb{E}_{\mathbb{T}}(X) = \dfrac{dC_f}{dz}(0),$$ | $$\mathbb{E}_{\mathbb{T}}(X) = \dfrac{dC_f}{dz}(0),$$ | ||
where $C_f$ is the [[cumulant generating function]] of $f$. | where $C_f$ is the [[cumulant generating function]] of $f$. | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\mathbb{E}_{\mathbb{T}}(X^k)=\displaystyle\int_0^{\infty} k! h_k(t,0)f(t) \Delta t.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
=References= | =References= | ||
[https://mospace.umsystem.edu/xmlui/bitstream/handle/10355/29595/Matthews_2011.pdf?sequence=1 Probability theory on time scales and applications to finance and inequalities by Thomas Matthews] | [https://mospace.umsystem.edu/xmlui/bitstream/handle/10355/29595/Matthews_2011.pdf?sequence=1 Probability theory on time scales and applications to finance and inequalities by Thomas Matthews] | ||
Revision as of 16:40, 14 April 2015
Let $\mathbb{T}$ be a time scale. Let $X$ be a random variable with probability density function $f \colon \mathbb{T} \rightarrow \mathbb{R}$. The expected value of $X$ is given by $$\mathbb{E}_{\mathbb{T}}(X) = \dfrac{dC_f}{dz}(0),$$ where $C_f$ is the cumulant generating function of $f$.
Properties
Theorem: The following formula holds: $$\mathbb{E}_{\mathbb{T}}(X^k)=\displaystyle\int_0^{\infty} k! h_k(t,0)f(t) \Delta t.$$
Proof: █
References
Probability theory on time scales and applications to finance and inequalities by Thomas Matthews
