Expected value

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