# 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: