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.

BREAKING DOWN '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 algebra 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.

A random variable can be either discrete or continuous. 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 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.

A random variable has a probability distribution which is 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 therefore 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.

Consider a probability distribution in which the outcomes of a random event are not equally likely to happen. If 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 4 different ways – TT, HT, TH, HH. Therefore, the P(Y=0) = ¼ 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 ¼. Notice that getting one head has a likelihood of occurring twice – HT and TH. In this case, P(Y=1) = 2/4 = ½.

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.

RELATED TERMS
  1. Random Walk Index

    Random Walk Index compares a security’s price movements to a ...
  2. Variable Death Benefit

    Variable death benefit refers to the amount paid out at death ...
  3. Variable Interest Rate

    A variable interest rate is a rate on a loan or security that ...
  4. Variability

    Variability is the extent to which data points in a statistical ...
  5. Variable Cost

    A variable cost is a corporate expense that changes in proportion ...
  6. Coefficient of Determination

    The coefficient of determination is a measure used in statistical ...
Related Articles
  1. Investing

    Scenario Analysis Provides Glimpse Of Portfolio Potential

    This statistical method estimates how far a stock might fall in a worst-case scenario.
  2. Investing

    What Are The Odds Of Scoring A Winning Trade?

    Just because you're on a winning streak doesn't mean you're a skilled trader. Find out why.
  3. Investing

    What's a Sensitivity Analysis?

    Sensitivity analysis is used in financial modeling to determine how one variable (the target variable) may be affected by changes in another variable (the input variable).
  4. Investing

    Stock and Flow Variables Explained: A Closer Look at Apple

    The difference between stock and flow variables is an essential concept in finance and economics. We illustrate with financial statements from Apple Inc.
  5. Investing

    Bet Smarter With the Monte Carlo Simulation

    This technique can reduce uncertainty in estimating future outcomes.
  6. Managing Wealth

    Variable Annuities: The Pros and Cons

    Variable annuities are one of the most complicated financial instruments—weighing the pros and cons.
  7. Investing

    How to use Monte Carlo simulation with GBM

    Learn how to estimate risk with the use of a Monte Carlo simulation to predict future events through a series of random trials.
  8. Investing

    What's the Correlation Coefficient?

    The correlation coefficient is a measure of how closely two variables move in relation to one another. If one variable goes up by a certain amount, the correlation coefficient indicates which ...
  9. Retirement

    Variable Annuities: The Do-It-Yourself Pension Plan

    Variable annuities can cost more than mutual funds, but that might be worth the protection they can add to your retirement.
RELATED FAQS
  1. What is the "random walk theory" and what does it mean for investors?

    The random walk theory is the occurrence of an event determined by a series of random movements - in other words, events ... Read Answer >>
  2. What are the advantages of using a simple random sample to study a larger population?

    Learn how simple random sampling works and what advantages it offers over other sampling methods when selecting a research ... Read Answer >>
  3. What is the difference between direct costs and variable costs?

    Learn about variable costs and direct costs, how direct costs and variable costs are classified and the differences between ... Read Answer >>
  4. When is it better to use systematic over simple random sampling?

    Learn when systematic sampling is better than simple random sampling, such as in the absence of data patterns and when there ... Read Answer >>
Hot Definitions
  1. Business Cycle

    The business cycle describes the rise and fall in production output of goods and services in an economy. Business cycles ...
  2. Futures Contract

    An agreement to buy or sell the underlying commodity or asset at a specific price at a future date.
  3. Yield Curve

    A yield curve is a line that plots the interest rates, at a set point in time, of bonds having equal credit quality, but ...
  4. Portfolio

    A portfolio is a grouping of financial assets such as stocks, bonds and cash equivalents, also their mutual, exchange-traded ...
  5. Gross Profit

    Gross profit is the profit a company makes after deducting the costs of making and selling its products, or the costs of ...
  6. Diversification

    Diversification is the strategy of investing in a variety of securities in order to lower the risk involved with putting ...
Trading Center