Difference between revisions of "Expected value"
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{{:Expected value of uniform distribution}} | {{:Expected value of uniform distribution}} | ||
{{:Expected value of exponential distribution}} | {{:Expected value of exponential distribution}} | ||
| + | {{:Expected value of gamma 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 22:14, 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 $f$.
Contents
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
Theorem
If $X$ is a random variable with the exponential distribution with parameter $\lambda$, then $$\mathrm{E}_{\mathbb{T}}(X)=\dfrac{1}{\lambda}.$$
Proof
References
Theorem: Let $X$ have the gamma distribution on $\mathbb{T}$. Then $$\mathrm{E}_{\mathbb{T}}(X)=\dfrac{k}{\lambda}.$$
Proof: █
References
Probability theory on time scales and applications to finance and inequalities by Thomas Matthews
