Difference between revisions of "Expected value"

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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
$$\mathbb{E}_{\mathbb{T}}(X) = \dfrac{dC_f}{dz}(0),$$
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$$\mathrm{E}_{\mathbb{T}}(X^k)=\displaystyle\int_{-\infty}^{\infty} k! h_k(t,0)f(t) \Delta t.$$
where $C_f$ is the [[cumulant generating function]] of $f$.
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=Properties=
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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<strong>Theorem:</strong> The following formula holds:
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$$\mathrm{E}_{\mathbb{T}}(X^k)=\displaystyle\int_{-\infty}^{\infty} k! h_k(t,0)f(t) \Delta t.$$
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<div class="mw-collapsible-content">
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<strong>Proof:</strong> █
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</div>
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</div>
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=Example=
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[[Expected value of uniform distribution]]<br />
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[[Expected value of exponential distribution]]<br />
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[[Expected value of gamma distribution]]<br />
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=References=
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[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]

Latest revision as of 15:57, 22 September 2016

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 $$\mathrm{E}_{\mathbb{T}}(X^k)=\displaystyle\int_{-\infty}^{\infty} k! h_k(t,0)f(t) \Delta t.$$

Properties

Theorem: The following formula holds: $$\mathrm{E}_{\mathbb{T}}(X^k)=\displaystyle\int_{-\infty}^{\infty} k! h_k(t,0)f(t) \Delta t.$$

Proof:

Example

Expected value of uniform distribution
Expected value of exponential distribution
Expected value of gamma distribution

References

Probability theory on time scales and applications to finance and inequalities by Thomas Matthews