Definition

1. Noun:
   • A variable quantity that is random: A numerical quantity whose value depends on the outcomes of a random phenomenon or experiment. It is a function that assigns a real number to each possible outcome in a sample space.

Usage Examples

• Noun:
  • The outcome of a coin flip can be modeled using a random variable. (Here, 'random variable' is the subject naming the mathematical concept used to represent the flip's result.)
  • We defined a random variable X to represent the sum of the two dice. (In this sentence, 'random variable' is the object being defined, specifying what 'X' is.)
  • The probability distribution describes the behavior of the random variable. (This example shows 'random variable' as the object of a preposition, indicating what the distribution describes.)

Advanced Usage

• "to be a function of a random variable": Indicates that one quantity's value is determined by the outcome of another random process.
  • The profit is a function of the random variable representing market demand.
• "independent and identically distributed (i.i.d.) random variables": A key concept in statistics referring to a sequence of random variables with the same probability distribution and mutual independence.
  • The theorem applies to a sequence of i.i.d. random variables.

Variants and Related Words

• Stochastic variable (n): A synonym for random variable, often used in more formal or theoretical contexts.
• Variate (n): Another less common synonym for random variable.
• Random vector (n): A generalization of a random variable to multiple dimensions (i.e., a vector whose components are random variables).

Synonyms

• Stochastic variable: A variable subject to randomness.
• Chance variable: Emphasizes the dependence on chance.

Related Phrases and Concepts

• Expected value of a random variable: The long-term average or mean value of the random variable.
  • Calculate the expected value of the random variable to find the average return.
• Variance of a random variable: A measure of the dispersion or spread of the random variable's possible values.
  • A high variance indicates the random variable's outcomes are widely spread.
• Realization of a random variable: The actual observed value that the random variable takes after an experiment is performed.
  • After rolling the die, the realization of the random variable was 4.

Noun

1. a variable quantity that is random