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

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