Definition:Random

Definition

Random Error

A random error is the discrepancy between an observation and a value predicted by a particular model.

Such an error represents uncontrolled variation.

Random errors are usually assumed to be independent with a normal distribution and zero expectation.

Random Sample (Probability Theory)

Let $X_i$ be a random variable with $\Img {X_i} = \Omega$, for all $1 \le i \le n$.

Let $F_i$ be the cumulative distribution function of $X_i$ for all $1 \le i \le n$.

We say that $X_1, X_2, \ldots, X_n$ form a random sample of size $n$ if:

$X_i$ and $X_j$ are independent if $i \ne j$

$\map {F_1} x = \map {F_i} x$ for all $x \in \Omega$

for all $1 \le i, j \le n$.

Random Sample (Statistics)

Let $P$ be a population.

A sample taken from $P$ is a random sample if and only if it is chosen in such a way that every possible sample of the same size has an equal probability of being selected.

Random Selection

A manner of selecting objects from some larger collection of objects is said to be random if the selection is made according to chance.

That is, there is no strict rule or procedure that predictably determines the outcome of the selection.

See experiment and random variable for a precise mathematical treatment of randomness.

Random Variable

A random variable is a number whose value is determined unambiguously by an experiment.

Random Vector

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $n \in \N$.

Let $\struct {S_1, \Sigma_1}$, $\struct {S_2, \Sigma_2}$, $\ldots$, $\struct {S_n, \Sigma_n}$ be measurable spaces.

Let:

$\ds S = \prod_{i \mathop = 1}^n S_i$

For each integer $1 \le i \le n$, let $X_i$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S_i, \Sigma_i}$.

Define a function $\mathbf X : \Omega \to S$ by:

$\map {\mathbf X} \omega = \tuple {\map {X_1} \omega, \map {X_2} \omega, \ldots, \map {X_n} \omega}$

for each $\omega \in \Omega$.

We call $\mathbf X$ a random vector.

One-Dimensional Random Walk

Let $\sequence {X_n}_{n \mathop \ge 0}$ be a Markov chain whose state space is the set of integers $\Z$.

Let $\sequence {X_n}$ be such that $X_{n + 1}$ is an element of the set $\set {X_n + 1, X_n, X_n - 1}$.

Then $\sequence {X_n}$ is a one-dimensional random walk.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): random
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): random