Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Further summary statistics ▷ Minimum

Definition: The minimum of a sample or random variable is its lowest observed or possible value.


1) Let $x = \left\lbrace x_1, \ldots, x_n \right\rbrace$ be a sample from a random variable $X$. Then, the minimum of $x$ is

\[\label{eq:min-samp} \mathrm{min}(x) = x_j, \quad \text{such that} \quad x_j \leq x_i \quad \text{for all} \quad i = 1, \ldots, n, \; i \neq j \; ,\]

i.e. the minimum is the value which is smaller than or equal to all other observed values.


2) Let $X$ be a random variable with possible values $\mathcal{X}$. Then, the minimum of $X$ is

\[\label{eq:min-rvar} \mathrm{min}(X) = \tilde{x}, \quad \text{such that} \quad \tilde{x} < x \quad \text{for all} \quad x \in \mathcal{X}\setminus\left\lbrace \tilde{x} \right\rbrace \; ,\]

i.e. the minimum is the value which is smaller than all other possible values.

 
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Metadata: ID: D107 | shortcut: min | author: JoramSoch | date: 2020-11-12, 05:25.