Variance

Let $\mathbb{T}$ be a time scale. Let $X$ be a random variable with probability density function $f \colon \mathbb{T} \rightarrow \mathbb{R}$. Then the variance of $X$ is defined by the formula $$\mathrm{Var}_{\mathbb{T}}(X) = \dfrac{d^2 C_f}{dz^2}(0).$$

Properties

Theorem: The following formula holds: $$\mathrm{Var}_{\mathbb{T}}(X) = \mathrm{E}_{\mathbb{T}}(X^2) - (\mathrm{E}_{\mathbb{T}}(X))^2.$$

Proof: █

Examples

Variance of uniform distribution
Variance of exponential distribution
Variance of gamma distribution

References

Probability theory on time scales and applications to finance and inequalities by Thomas Matthews