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 | ||
| − | $$\mathrm{E}_{\mathbb{T}}(X) = \ | + | $$\mathrm{E}_{\mathbb{T}}(X^k)=\displaystyle\int_{-\infty}^{\infty} k! h_k(t,0)f(t) \Delta t.$$ |
| − | |||
=Properties= | =Properties= | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
<strong>Theorem:</strong> The following formula holds: | <strong>Theorem:</strong> The following formula holds: | ||
| − | $$\mathrm{E}_{\mathbb{T}}(X^k)=\displaystyle\ | + | $$\mathrm{E}_{\mathbb{T}}(X^k)=\displaystyle\int_{-\infty}^{\infty} k! h_k(t,0)f(t) \Delta t.$$ |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
| Line 13: | Line 12: | ||
=Example= | =Example= | ||
| − | + | [[Expected value of uniform distribution]]<br /> | |
| − | + | [[Expected value of exponential distribution]]<br /> | |
| + | [[Expected value of gamma distribution]]<br /> | ||
=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] | ||
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
