Difference between revisions of "Expected value"
From timescalewiki
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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 | 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 | ||
| − | $$\ | + | $$\mathrm{E}_{\mathbb{T}}(X) = \dfrac{dC_f}{dz}(0),$$ |
where $C_f$ is the [[cumulant generating function]] of $f$. | where $C_f$ is the [[cumulant generating function]] of $f$. | ||
| Line 6: | Line 6: | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
<strong>Theorem:</strong> The following formula holds: | <strong>Theorem:</strong> The following formula holds: | ||
| − | $$\ | + | $$\mathrm{E}_{\mathbb{T}}(X^k)=\displaystyle\int_0^{\infty} k! h_k(t,0)f(t) \Delta t.$$ |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
</div> | </div> | ||
</div> | </div> | ||
| + | |||
| + | =Example= | ||
| + | {{:Expected value of uniform distribution}} | ||
=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 22:01, 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 $$\mathrm{E}_{\mathbb{T}}(X) = \dfrac{dC_f}{dz}(0),$$ where $C_f$ is the cumulant generating function of $f$.
Properties
Theorem: The following formula holds: $$\mathrm{E}_{\mathbb{T}}(X^k)=\displaystyle\int_0^{\infty} k! h_k(t,0)f(t) \Delta t.$$
Proof: █
Example
Theorem
Let $X$ have the uniform distribution on $[a,b] \cap \mathbb{T}$. Then, $$\mathrm{E}_{\mathbb{T}}(X) = \dfrac{h_2(\sigma(b),a)}{\sigma(b)-a}+a,$$ where $h_2$ denotes the delta hk and $\sigma$ denotes the forward jump.
Proof
References
References
Probability theory on time scales and applications to finance and inequalities by Thomas Matthews
