Two Types of Random Variables
A random variable
Learning Objective

Contrast discrete and continuous variables
Key Points
 A random variable is a variable taking on numerical values determined by the outcome of a random phenomenon.
 The probability distribution of a random variable
$x$ tells us what the possible values of$x$ are and what probabilities are assigned to those values.  A discrete random variable has a countable number of possible values.
 The probability of each value of a discrete random variable is between 0 and 1, and the sum of all the probabilities is equal to 1.
 A continuous random variable takes on all the values in some interval of numbers.
 A density curve describes the probability distribution of a continuous random variable, and the probability of a range of events is found by taking the area under the curve.
Terms

discrete random variable
obtained by counting values for which there are no inbetween values, such as the integers 0, 1, 2, ….

continuous random variable
obtained from data that can take infinitely many values

random variable
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
Full Text
Random Variables
In probability and statistics, a randomvariable is a variable whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense). 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.
A random variable's possible values might represent the possible outcomes of a yettobeperformed experiment, or the possible outcomes of a past experiment whose alreadyexisting value is uncertain (for example, as a result of incomplete information or imprecise measurements). They may also conceptually represent either the results of an "objectively" random process (such as rolling a die), or the "subjective" randomness that results from incomplete knowledge of a quantity.
Random variables can be classified as either discrete (that is, taking any of a specified list of exact values) or as continuous (taking any numerical value in an interval or collection of intervals). The mathematical function describing the possible values of a random variable and their associated probabilities is known as a probability distribution.
Discrete Random Variables
Discrete random variables can take on either a finite or at most a countably infinite set of discrete values (for example, the integers). Their probability distribution is given by a probability mass function which directly maps each value of the random variable to a probability. For example, the value of
Discrete Probability Disrtibution
This shows the probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.
Examples of discrete random variables include the values obtained from rolling a die and the grades received on a test out of 100.
Continuous Random Variables
Continuous random variables, on the other hand, take on values that vary continuously within one or more real intervals, and have a cumulative distribution function (CDF) that is absolutely continuous. As a result, the random variable has an uncountable infinite number of possible values, all of which have probability 0, though ranges of such values can have nonzero probability. The resulting probability distribution of the random variable can be described by a probability density, where the probability is found by taking the area under the curve.
Probability Density Function
The image shows the probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important continuous random distribution. As notated on the figure, the probabilities of intervals of values corresponds to the area under the curve.
Selecting random numbers between 0 and 1 are examples of continuous random variables because there are an infinite number of possibilities.
Key Term Reference
 cumulative distribution function
 Appears in these related concepts: Continuous Probability Distributions, The Uniform Distribution, and Optional Collaborative Classrom Activity
 density
 Appears in these related concepts: Density Calculations, Volume and Density, and The Density Scale
 distribution
 Appears in these related concepts: Application of Knowledge, Monte Carlo Simulation, and Selling to Consumers
 experiment
 Appears in these related concepts: Fundamentals of Probability, Descriptive and Correlational Statistics, and Primary Market Research
 finite
 Appears in these related concepts: The Sample Average, Introduction to Sequences, and Summing Terms in an Arithmetic Sequence
 probability
 Appears in these related concepts: Theoretical Probability, Rules of Probability for Mendelian Inheritance, and The Addition Rule
 probability distribution
 Appears in these related concepts: Probability Distributions for Discrete Random Variables, Recognizing and Using a Histogram, and Overview of How to Assess StandAlone Risk
 probability mass function
 Appears in these related concepts: The Binomial Formula and The Hypergeometric Random Variable
 random number
 Appears in these related concepts: Random Samples, Are Real Dice Fair?, and Lab 2: Central Limit Theorem (Cookie Recipes)
 range
 Appears in these related concepts: The Derivative as a Function, Visualizing Domain and Range, and Introduction to Domain and Range
 statistics
 Appears in these related concepts: Communicating Statistics, Basic Inferential Statistics, and Understanding Statistics
 variable
 Appears in these related concepts: What is a Linear Function?, Math Review, and Introduction to Variables
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