Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability distributions ▷ Scale parameter

Definition: Let $\mathcal{D}(\lambda, \sigma)$ denote a family of probability distributions with statistical parameters $\lambda$ and $\sigma$. Then, the parameter $\sigma$ is referred to as a “scale parameter”, if and only if the following holds: When a random variable $X$ is following the probability distribution $\mathcal{D}(\lambda^{*}, \sigma^{*})$, then the random variable $Y = X/\sigma^{*}$ is following the probability distribution $\mathcal{D}(\lambda^{*}, 1)$.

\[\label{eq:para-scal} X \sim \mathcal{D}(\lambda^{*}, \sigma^{*}) \quad \Rightarrow \quad Y = \frac{X}{\sigma^{*}} \sim \mathcal{D}(\lambda^{*}, 1) \; .\]

This implies an identity of the cumulative distribution functions:

\[\label{eq:para-scal-cdf} \begin{split} F_Y(y) &= F_X(\sigma^{*} y) \\ F_\mathcal{D}(y; \, \lambda^{*}, 1) &= F_\mathcal{D}(\sigma^{*} y; \, \lambda^{*}, \sigma^{*}) \; . \end{split}\]

The parameter $\sigma$ can be scalar for a random variable $X$ or matrix-valued for a random matrix $X$.

 
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Metadata: ID: D218 | shortcut: para-scal | author: JoramSoch | date: 2025-03-14, 15:02.