Random Variables

Random variables refer to unknown values or functions that help determine an event's probability by assigning a quantity to the outcome. Simply, it denotes those variables occupying a random experiment's sample space. These independent random variables can be discrete or continuous based on the range of values they can take.

Random variables are frequently used in diverse fields like science, economics, and finance. For instance, in finance, it is used in risk analysis and management. In addition, businesses often use these variables to determine the return on investment. Finally, governments use such variables to estimate an event's occurrence or lack thereof.

• Random variables in statistics are unknown values or functions which can serve as input to determine the probability of an event.
• First, one must determine the sample space and the favorable outcomes to find the probability distribution. Then, the variables of a random experiment occupy the sample space.
• Its functions can help find the expected value of a probability distribution for discrete and continuous variables.
• Selecting investments based on ROI and the risk involved is extremely helpful.

Random variables can be understood as the most basic elements of statistical probability. While calculating the likelihood of any event, the possible values which could lead to a certain outcome are prerequisites. These values are the inputs present during a random experiment.

In finance and statistical analysis, random variables are fundamental concepts used to model uncertain outcomes or events. A random variable represents a numerical quantity whose value is subject to variation due to chance or randomness. These variables are integral to understanding and quantifying uncertainty in various financial scenarios.

Random variables statistics can be categorized as either discrete or continuous. Discrete random variables take on specific, distinct values, often associated with counting or categorizing outcomes. For instance, the number of heads in a series of coin tosses is a discrete random variable. On the other hand, continuous random variables can take any value within a given range and are associated with measurements rather than counts, such as stock prices or interest rates.

The probability distribution of a random variable describes the likelihood of different outcomes occurring. In finance, understanding the probability distribution of variables like asset returns is crucial for risk management and portfolio optimization.

By incorporating them into financial models, analysts can simulate and analyze the range of potential outcomes and associated probabilities. This enables a more comprehensive understanding of risk, aiding investors and decision-makers in making wiser choices and developing strategies that account for the inherent uncertainty in financial markets.

Though it might seem simple, the concept finds a wide range of applications in many fields. It is most commonly popular in risk management, as it helps determine the possibility of a high-risk event. In addition, companies and investors use random variables to calculate the returns on investment and the associated payback period.

There are two types of random variable statistics that are most commonly used in random experiments. Let us understand them through the discussion below. 

#1 – Discrete random variables

These variables can take only finite, countable values in the discrete probability distribution. Therefore, only positive, non-decimal, and whole numbers can be the input values to calculate the likelihood of a certain outcome.

For example, when a person tosses a coin and considers the number of times tails can come up, it will either be 0, 1, or 2. The probability of an event using discrete variables can be determined using binomial, multinomial, Bernoulli, and Poisson distributions.

#2 – Continuous random variables

Continuous variables find the probability of any value, from negative to positive infinity. That is, the values can also be negative, decimals or fractions. This can help analyze a complex set of data. For example, if a person sets to find the exact heights of people worldwide, they would get many different decimal values. 

The area under a density curve often represents continuous curves, implying that a continuum of values in specified intervals can belong to the sample space of an event.

Independent random variable functions enable the calculation of expectations or expected values. Expectations refer to the sum of probabilities of all the possible outcomes. For example, in the case of throwing a die, it is 1/6 x 6 = 1. If throwing a die and getting an even number, it is 1/6 x 3 = ½.

Suppose Y is a random variable and g(X) is a real function for all values of X. Then, the cumulative distribution function (CDF) of Y can be represented as:

The cumulative distribution function shows the overall distribution of variables. It determines all the values of a function when X will take a value less than or equal to y, i.e., the favorable outcomes.

Now, if X is a discrete variable,

Here, SX is the support of X or the set of all the values in the domain that are not mapped to zero in the range. PX is the probability mass function of X.

If X is a continuous variable,

Here, FX is the probability distribution function of X.

Now that we understand the basics of random variables statistics in detail, it is only just that we also understand its practical applicability. Let us do so through the examples below.

Example #1

Consider a simple experiment where a person throws two dies simultaneously. A person wants to find the number of possibilities when both the die shows an odd prime number. Here, the random variables include all the possibilities that could come up when two dies are thrown.

Sample space, S = { (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) }

The possible outcomes, as per the desired event, E = { (3, 3), (3, 5), (5, 3), (5, 5) }

Probability of the event, P (E) = n (E)/ n (S)

= 4/ 36 = 1/ 9

Example #2

Recently, Forbes published an article stating that statistical literacy would help advance the role of artificial intelligence in modernizing business. This is because business is all about data which requires statistical analysis to be transformed into a more usable form. In addition, any statistical analysis needs the use of random variables for its effective execution.

These variables are critical for various statistical analytics tools like A/B testing, correlation and regression analysis, clustering, causal interference, cross-validation, hypothesis testing, standard error determination, and population analysis.

Random variable problems often arise in finance and statistics, posing challenges that require analytical solutions to navigate uncertainty and assess risk. Some key issues associated with independent random variables include:

• Random variables represent uncertain quantities, making it challenging to predict precise outcomes. The inherent variability in financial markets necessitates robust models that can effectively capture and quantify this uncertainty.
• Determining the probability distribution of a random variable is essential for risk assessment. Estimating accurate distribution parameters is a common problem, especially when dealing with real-world financial data.
• Simulating the behavior of random variables in complex financial systems requires sophisticated techniques. Monte Carlo simulations, for example, involve generating numerous random scenarios to model the range of possible outcomes and associated probabilities.
• Understanding extreme events or tail risks is critical for risk management. Random variable problems often involve assessing the likelihood and potential impact of rare but significant market events.
• Random variables in financial models are rarely independent. Analyzing and modeling the correlation and dependence between variables is essential for constructing accurate risk assessments and portfolio strategies.
• Financial time series data involves the analysis of random variables over time. Identifying patterns, trends, and potential dependencies in these time series is crucial for making informed predictions and decisions.

This has been a guide to What are Random Variables. Here, we explain its types and functions, examples, and problems in detail. You may also find some useful articles here:

• Binomial Distribution
• Multinomial Distribution
• Probability Density Function