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# 4.1: Two Types of Random Variables

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Learn to distinguish between the two types of random variables: continuous and discrete.

The word discrete means countable. For example, the number of students in a class is countable or discrete. The value could be 2,24,34,\begin{align*}2, 24, 34,\end{align*} or 135\begin{align*}135\end{align*} students but it cannot be 232/3\begin{align*}23 2/3\end{align*} or 12.23\begin{align*}12.23\end{align*} students. The cost of a loaf of bread is also discrete, say \$3.17\begin{align*}\3.17\end{align*}, where we are counting dollars and cents, but not fractions of a cent.

However, if we are measuring the tire pressure in an automobile, we are dealing with a continuous variable. The air pressure can take values from 0\begin{align*}0\end{align*} psi to some large amount that would cause the tire to burst. Another example is the height of your fellow students in your classroom. The values could be anywhere from, say, 4.5feet\begin{align*}4.5 \;\mathrm{feet}\end{align*} to 7.2feet\begin{align*}7.2 \;\mathrm{feet}\end{align*}. In general, quantities such as pressure, height, mass, weight, density, volume, temperature, and distance are examples of continuous variables. Discrete random variables come usually from counting, say, the number of chickens in a coop, or the number of passing scores on an exam or the number of voters who showed up to the polls.

One way of distinguishing discrete and continuous variables is between any two values of a continuous variable, there are an infinite number of other valid values. This is not the case for discrete variables; between any two discrete values, there are an integer number (0,1,2,)\begin{align*}(0, 1, 2, \ldots)\end{align*} of valid values. For a discrete variable, these are considered countable values since you could count a whole number of them.

## Discrete Random Variables and Continuous Random Variables

Random variables that assume a countable number of values are called discrete.

Random variables that can take any of the countless number of values are called continuous.

Example:

Here is a list of discrete random variables:

1. The number of cars sold by a car dealer in one month: x=0,1,2,3,\begin{align*}x = 0, 1, 2, 3, \ldots\end{align*}
2. The number of students who were protesting the tuition increase last semester: x=0,1,2,3,.\begin{align*}x = 0, 1, 2, 3, \ldots.\end{align*} Notice that x\begin{align*}x\end{align*} could become very large.
3. The number of applicants who have applied for the vacant position at a company: x=0,1,2,3,\begin{align*}x = 0, 1, 2, 3, \ldots\end{align*}
4. The number of typographical errors in a rough draft of a book: x=0,1,2,3,\begin{align*}x = 0, 1, 2, 3, \ldots\end{align*}

Example:

Here is a list of continuous random variables:

1. The length of time it took the truck driver to go from New York city to Miami: x>0\begin{align*}x > 0\end{align*}, where x\begin{align*}x\end{align*} is the time.
2. The depth of oil drilling to find oil: 0<x<c\begin{align*}0 < x < c\end{align*}, where c\begin{align*}c\end{align*} is the maximum depth possible.
3. The weight of a truck in a truck weighing station: 0<x<c\begin{align*}0 < x < c\end{align*}, where c\begin{align*}c\end{align*} is the maximum weight possible.
4. The amount of water loaded in a 12ounce\begin{align*}12-\;\mathrm{ounce}\end{align*} bottle in a bottle filling operation: 0<x<12.\begin{align*}0 < x < 12.\end{align*}

## Lesson Summary

1. A random variable represents the numerical value of a simple event of an experiment.
2. Random variables that assume a countable number of values are called discrete.
3. Random variables that can take any of the countless number of values are called continuous.

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