random variable

(noun)

a quantity whose value is random and to which a probability distribution is assigned, such as the possible outcome of a roll of a die

Related Terms

  • random walk
  • stochastic
  • continuous random variable
  • discrete random variable
  • finite
  • weighted average
  • integral

Examples of random variable in the following topics:

  • Introduction

    • Continuous random variables have many applications.
    • The field of reliability depends on a variety of continuous random variables.
    • This chapter gives an introduction to continuous random variables and the many continuous distributions.
    • NOTE: The values of discrete and continuous random variables can be ambiguous.
    • How the random variable is defined is very important.
  • Two Types of Random Variables

    • A random variable $x$, and its distribution, can be discrete or continuous.
    • As opposed to other mathematical variables, a random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a set of possible different values, each with an associated probability.
    • Discrete random variables can take on either a finite or at most a countably infinite set of discrete values (for example, the integers).
    • Examples of discrete random variables include the values obtained from rolling a die and the grades received on a test out of 100.
    • Selecting random numbers between 0 and 1 are examples of continuous random variables because there are an infinite number of possibilities.
  • Expected Values of Discrete Random Variables

    • The expected value of a random variable is the weighted average of all possible values that this random variable can take on.
    • A discrete random variable $X$ has a countable number of possible values.
    • The probability distribution of a discrete random variable $X$ lists the values and their probabilities, such that $x_i$ has a probability of $p_i$.
    • In probability theory, the expected value (or expectation, mathematical expectation, EV, mean, or first moment) of a random variable is the weighted average of all possible values that this random variable can take on.
    • The weights used in computing this average are probabilities in the case of a discrete random variable.
  • Random Variable Notation

    • Upper case letters like X or Y denote a random variable.
    • Lower case letters like x or y denote the value of a random variable.
    • If X is a random variable, then X is written in words. and x is given as a number.
    • Because you can count the possible values that X can take on and the outcomes are random (the x values 0, 1, 2, 3), X is a discrete random variable.
  • Probability Distributions for Discrete Random Variables

    • Probability distributions for discrete random variables can be displayed as a formula, in a table, or in a graph.
    • A discrete random variable $x$ has a countable number of possible values.
    • The probability mass function has the same purpose as the probability histogram, and displays specific probabilities for each discrete random variable.
    • This histogram displays the probabilities of each of the three discrete random variables.
    • This table shows the values of the discrete random variable can take on and their corresponding probabilities.
  • The Central Limit Theorem for Sums

    • Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution) and suppose:
    • If you draw random samples of size n, then as n increases, the random variable ΣX which consists of sums tends to be normally distributed and
  • The Hypergeometric Random Variable

    • A hypergeometric random variable is a discrete random variable characterized by a fixed number of trials with differing probabilities of success.
    • As random selections are made from the population, each subsequent draw decreases the population causing the probability of success to change with each draw.
    • A random variable follows the hypergeometric distribution if its probability mass function is given by:
  • Bernoulli distribution

    • A Bernoulli random variable has exactly two possible outcomes.
    • Bernoulli random variables are often denoted as 1 for a success and 0 for a failure.
    • This mathematical inquiry of Bernoulli random variables can be extended even further.
    • If X is a random variable that takes value 1 with probability of success p and 0 with probability 1−p, then X is a Bernoulli random variable with mean and standard deviation µ = p, σ =$\sqrt{p(1-p)}$
    • In general, it is useful to think about a Bernoulli random variable as a random process with only two outcomes: a success or failure.
  • Chance Processes

    • A stochastic process is a collection of random variables that is often used to represent the evolution of some random value over time.
    • In probability theory, a stochastic process--sometimes called a random process-- is a collection of random variables that is often used to represent the evolution of some random value, or system, over time.
    • Random variables are non-deterministic (single) quantities which have certain probability distributions.
    • Random variables corresponding to various times (or points, in the case of random fields) may be completely different.
    • Although the random values of a stochastic process at different times may be independent random variables, in most commonly considered situations they exhibit complicated statistical correlations.
  • Randomized Design: Single-Factor

    • Completely randomized designs study the effects of one primary factor without the need to take other nuisance variables into account.
    • In the design of experiments, completely randomized designs are for studying the effects of one primary factor without the need to take into account other nuisance variables.
    • The experiment under a completely randomized design compares the values of a response variable based on the different levels of that primary factor.
    • An example of a completely randomized design using the three numbers is:
    • Discover how randomized experimental design allows researchers to study the effects of a single factor without taking into account other nuisance variables.
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.