What Are Random Variables?

You are free to use this image on you website, templates, etc., Please provide us with an attribution linkHow to Provide Attribution?Article Link to be HyperlinkedFor eg:Source: Random Variables (wallstreetmojo.com)

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.

Key Takeaways

  • 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 in Statistics Explained

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.

Let’s understand this concept by examining a person drawing cards from a deck. If they draw out a black card, the person loses. But, on the other hand, if they draw out a red card, they win. Since the number of black and red cards is equal in a deck, the probability of the person winning will be ½.

The real possibilities here are the total number of cards, which is 52. The favorable outcomes (possibilities where the person wins = number of red cards) = 26. In this case, 52 cards are the random variables.

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.

Types

There are two types of variables that are common in random experiments. 

#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.

Functions

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.

Examples of Random Variables

Here are some examples to understand the variables involved in random experiments.

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.

This has been a guide to What is Random Variables and its definition. Here, we explain its types and functions along with examples. You may also find some useful articles here:

Discrete variables are those which have distinct and finite values. Hence, only positive, whole numbers can be acceptable as discrete variables. Therefore, it is appropriate for analyzing simple datasets.

Continuous variables are the opposite of discrete variables. They can take any values, negatives, decimals, rational numbers, etc. Hence, the continuum of data is under the density curve. Therefore, it is most suitable for complex sets of data.

Random variables can take up the values that determine the probability of a particular outcome in an event. It usually occupies the sample space of an event. Sample space is the set of all possibilities for a particular event, favorable or not.

To find the probability of a particular outcome, the random variables must be input and the probability determined.

  • Binomial DistributionMultinomial DistributionProbability Density Function