Random variable

Random variable

A random variable is a variable whose possible values are outcomes of a random phenomenon. More specifically, a random variable is defined as a function that maps the outcomes of an unpredictable process to numerical quantities, typically real numbers, which can be either discrete (i.e., taking any of a specified list of exact values) or continuous (i.e., taking any numerical value in an interval or collection of intervals).

Definition

Formally, let S be a sample space — a set of outcomes of a random process. Then a random variable X is a function from S to the real numbers. The technical conditions on the function X are designed to capture the idea that the value of X is determined by the outcome of the random process.

Types of random variables

Random variables can be classified into two broad types:

• Discrete random variables: These are variables that can take on a countable number of values. Examples include the number of heads in a series of coin flips, or the number of students present in a classroom.

• Continuous random variables: These are variables that can take on any value in a given range or interval. Examples include the height of a person, or the time it takes for a computer to process a task.

Probability distribution

The Probability distribution of a random variable is a function that describes the likelihood of each possible outcome. It is defined for every possible outcome in the sample space and satisfies the conditions of a probability measure.

Expectation and variance

The Expected value or expectation of a random variable is a weighted average of all possible values that the random variable can take on, with the weights being the probabilities of these outcomes. The Variance of a random variable is a measure of how much values of the random variable vary around the expected value.

See also

• Probability theory
• Statistics
• Stochastic process
• Probability density function
• Cumulative distribution function

References

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