Difference between revisions of "Variance"
From timescalewiki
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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 | 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 | ||
$$\mathbb{V}ar_{\mathbb{T}}(X) = \dfrac{d^2 C_f}{dz^2}(0).$$ | $$\mathbb{V}ar_{\mathbb{T}}(X) = \dfrac{d^2 C_f}{dz^2}(0).$$ | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\mathbb{V}ar_{\mathbb{T}}(X) = \mathbb{E}_{\mathbb{T}}(X^2) - (\mathbb{E}_{\mathbb{T}}(X))^2.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> proof goes here █ | ||
| + | </div> | ||
| + | </div> | ||
Revision as of 17:24, 23 November 2014
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 $$\mathbb{V}ar_{\mathbb{T}}(X) = \dfrac{d^2 C_f}{dz^2}(0).$$
Properties
Theorem: The following formula holds: $$\mathbb{V}ar_{\mathbb{T}}(X) = \mathbb{E}_{\mathbb{T}}(X^2) - (\mathbb{E}_{\mathbb{T}}(X))^2.$$
Proof: proof goes here █
