## What Is a Random Variable?

A random variable is a variable whose value is unknown or a function that assigns values to each of an experiment's outcomes. Random variables are often designated by letters and can be classified as discrete, which are variables that have specific values, or continuous, which are variables that can have any values within a continuous range.

Random variables are often used in econometric or regression analysis to determine statistical relationships among one another.

### Key Takeaways

- A random variable is a variable whose value is unknown or a function that assigns values to each of an experiment's outcomes.
- A random variable can be either discrete (having specific values) or continuous (any value in a continuous range).
- The use of random variables is most common in probability and statistics, where they are used to quantify outcomes of random occurrences.
- Risk analysts use random variables to estimate the probability of an adverse event occurring.

#### What is a Random Variable?

## Understanding a Random Variable

In probability and statistics, random variables are used to quantify outcomes of a random occurrence, and therefore, can take on many values. Random variables are required to be measurable and are typically real numbers. For example, the letter X may be designated to represent the sum of the resulting numbers after three dice are rolled. In this case, X could be 3 (1 + 1+ 1), 18 (6 + 6 + 6), or somewhere between 3 and 18, since the highest number of a die is 6 and the lowest number is 1.

A random variable is different from an algebraic variable. The variable in an algebraic equation is an unknown value that can be calculated. The equation 10 + x = 13 shows that we can calculate the specific value for x which is 3. On the other hand, a random variable has a set of values, and any of those values could be the resulting outcome as seen in the example of the dice above.

In the corporate world, random variables can be assigned to properties such as the average price of an asset over a given time period, the return on investment after a specified number of years, the estimated turnover rate at a company within the following six months, etc. Risk analysts assign random variables to risk models when they want to estimate the probability of an adverse event occurring. These variables are presented using tools such as scenario and sensitivity analysis tables which risk managers use to make decisions concerning risk mitigation.

## Types of Random Variables

A random variable has a probability distribution that represents the likelihood that any of the possible values would occur. Let’s say that the random variable, Z, is the number on the top face of a die when it is rolled once. The possible values for Z will thus be 1, 2, 3, 4, 5, and 6. The probability of each of these values is 1/6 as they are all equally likely to be the value of Z.

For instance, the probability of getting a 3, or P (Z=3), when a die is thrown is 1/6, and so is the probability of having a 4 or a 2 or any other number on all six faces of a die. Note that the sum of all probabilities is 1.

A random variable can be either discrete or continuous.

### Discrete Random Variables

Discrete random variables take on a countable number of distinct values. Consider an experiment where a coin is tossed three times. If X represents the number of times that the coin comes up heads, then X is a discrete random variable that can only have the values 0, 1, 2, 3 (from no heads in three successive coin tosses to all heads). No other value is possible for X.

### Continuous Random Variables

Continuous random variables can represent any value within a specified range or interval and can take on an infinite number of possible values. An example of a continuous random variable would be an experiment that involves measuring the amount of rainfall in a city over a year or the average height of a random group of 25 people.

Drawing on the latter, if Y represents the random variable for the average height of a random group of 25 people, you will find that the resulting outcome is a continuous figure since height may be 5 ft or 5.01 ft or 5.0001 ft. Clearly, there is an infinite number of possible values for height.

## Example of a Random Variable

A typical example of a random variable is the outcome of a coin toss. Consider a probability distribution in which the outcomes of a random event are not equally likely to happen. If the random variable Y is the number of heads we get from tossing two coins, then Y could be 0, 1, or 2. This means that we could have no heads, one head, or both heads on a two-coin toss.

However, the two coins land in four different ways: TT, HT, TH, and HH. Therefore, the P(Y=0) = 1/4 since we have one chance of getting no heads (i.e., two tails [TT] when the coins are tossed). Similarly, the probability of getting two heads (HH) is also 1/4. Notice that getting one head has a likelihood of occurring twice: in HT and TH. In this case, P (Y=1) = 2/4 = 1/2.

## What Is a Discrete Random Variable?

A discrete random variable is a type of random variable that has a countable number of distinct values that can be assigned to it, such as in a coin toss.

## What Is a Continuous Random Variable?

A continuous random variable stands for any amount within a specific range or set of points and can reflect an infinite number of potential values, such as the average rainfall in a region.

## What Is a Mixed Random Variable?

A mixed random variable combines elements of both discrete and continuous random variables.

## How Do You Identify a Random Variable?

A random variable is one whose value is unknown a priori, or else is assigned a random value based on some data generating process or mathematical function.

## Why Are Random Variables Important?

Random variables produce probability distributions based on experimentation, observation, or some other data-generating process. Random variables, in this way, allow us to understand the world around us based on a sample of data, by knowing the likelihood that a specific value will occur in the real world or at some point in the future.

## The Bottom Line

Random variables, whether discrete or continuous, are a key concept in statistics and experimentation. Because they are random with unknown exact values, these allow us to understand the probability distribution of those values or the relative likelihood of certain events. As a result, analysts can test hypotheses and make inferences about the natural and social world around us.