discrete variable
a variable that takes values from a finite or countable set, such as the number of legs of an animal
Examples of discrete variable in the following topics:

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.

Types of Variables
 Numeric variables may be further described as either continuous or discrete.
 A discrete variable is a numeric variable.
 A discrete variable cannot take the value of a fraction between one value and the next closest value.
 Variables can be numeric or categorial, being further broken down in continuous and discrete, and nominal and ordinal variables.
 Distinguish between quantitative and categorical, continuous and discrete, and ordinal and nominal variables.

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.

Two Types of Random Variables
 A random variable $x$, and its distribution, can be discrete or continuous.
 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).
 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.
 This shows the probability mass function of a discrete probability distribution.

Variables
 In this case, the variable is "type of antidepressant. " When a variable is manipulated by an experimenter, it is called an independent variable.
 An important distinction between variables is between qualitative variables and quantitative variables.
 Qualitative variables are sometimes referred to as categorical variables.
 Variables such as number of children in a household are called discrete variables since the possible scores are discrete points on the scale.
 Other variables such as "time to respond to a question" are continuous variables since the scale is continuous and not made up of discrete steps.

Probability Distribution Function (PDF) for a Discrete Random Variable
 This is a discrete PDF because

Introduction
 These two examples illustrate two different types of probability problems involving discrete random variables.
 Recall that discrete data are data that you can count.
 A random variable describes the outcomes of a statistical experiment in words.
 The values of a random variable can vary with each repetition of an experiment.
 In this chapter, you will study probability problems involving discrete random distributions.

Introduction
 Continuous random variables have many applications.
 The ﬁeld of reliability depends on a variety of continuous random variables.
 NOTE: The values of discrete and continuous random variables can be ambiguous.
 For example, if X is equal to the number of miles (to the nearest mile) you drive to work, then X is a discrete random variable.
 How the random variable is deﬁned is very important.

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.
 The hypergeometric distribution is a discrete probability distribution that describes the probability of $k$ successes in $n$ draws without replacement from a finite population of size $N$ containing a maximum of $K$ successes.
 A random variable follows the hypergeometric distribution if its probability mass function is given by:

Types of variables
 For this reason, the population variable is said to be discrete since it can only take numerical values with jumps.
 A variable with these properties is called an ordinal variable.
 To simplify analyses, any ordinal variables in this book will be treated as categorical variables.
 Classify each of the variables as continuous numerical, discrete numerical, or categorical.
 Because the number of siblings is a count, it is discrete.