Transformation of Variables

A random variable is a real-valued function defined over the sample space of a random experiment in probability. That is, the random variable’s values correlate to the results of the random experiment. Random variables can be discrete or continuous in nature.

The likely values of a random variable may represent the possible results of an experiment that is about to be conducted or the possible outcomes of a previous experiment whose value is unknown. They can also be used to explain the outcomes of a “objectively” random process (such as rolling a die) or the “subjective” randomness that arises from a lack of understanding of a quantity.

A sample space is the collection of alternative outcomes of a random event, and it is the domain of a random variable. When a coin is tossed, for example, there are only two possible outcomes: heads or tails.

Definition of Random Variables

A random variable is a rule that gives each outcome in a sample space a numerical value. Random variables can be discrete or continuous in nature. If a random variable adopts only specified values in an interval, it is said to be discrete. It is continuous otherwise. Random variables are usually denoted by capital letters, such as X and Y. A discrete random variable is one that takes the values  1, 2, 3,…

A random variable must be measured as a function, allowing probabilities to be given to a set of probable values. The outcomes are obviously dependent on some physical elements that are unpredictable. For example, if we throw a fair coin, the final result of heads or tails will be determined by the physical conditions. We have no way of knowing which outcome will be remembered. Other possibilities, such as the coin breaking or becoming lost, are not taken into account.

Variate

A variate is simply a generalisation of a random variable. It has the same qualities as random variables without being restricted to any certain sort of probabilistic experiment. It always follows a certain probability rule.

When a variate cannot assume all of the values in the given range, it is referred to as a discrete variate.

The term “continuous variate” refers to a variate that can assume all of the numerical values in a given range.

Types of Random Variables

There are two random variables, as mentioned in the introduction:

• Discrete Random Variable

• Continuous Random Variable

Let’s take a closer look at various types of variables, along with examples.

Discrete Random Variable

A discrete random variable can only have a limited number of different values, such as 0, 1, 2, 3, 4, etc. The probability mass function is a list of probabilities compared to each of the possible values in a random variable’s probability distribution.

Allow a person to be chosen at random in an analysis, and a random variable to represent the person’s height. The random variable is logically described as a function that connects the individual to their height. In terms of the random variable, it is a probability distribution that allows the calculation of the probability that the height falls into any subset of likely values, such as the likelihood that the height falls between 175 and 185 cm, or the possibility that the height falls between 145 and 180 cm. Another random variable is the person’s age, which could range from 45 to 50 years old, could be less than 40 or greater than 50.

Continuous Random Variable

A numerically valued variable is considered to be continuous if it can take on the values a and b in any unit of measurement. The random variable X is considered to be continuous if it can take on an unlimited and uncountable set of values. X is considered to be a continuous random variable in that interval when it takes any value in the interval (a, b).

A continuous random variable is one with a constant cumulative distribution function throughout. There are no “gaps” between the numbers that may be compared to numbers with a low likelihood of appearing. Alternatively, these variables nearly never assume an exactly defined value c, but there is a positive likelihood that their value will rest in small intervals.

Transformation of Random Variables

A random variable’s value is reassigned to another variable when it is transformed. After inserting the transformation to remap the number line from x to y, the transformation function is y = g. (x).

Transformation of X or Expected Value of X for a Continuous Variable

If the random variable X has the values x1, x2, x3, ….. etc., and have the probability as P (x1), P (x2), P (x3), …… etc., then the random variable’s expected value is given by

X’s expectation, E (x) =ƒ x P (x)

Conclusion

A random variable is a variable whose value is unknown or a function whose values are determined by the results of a random experiment. Random variables have different letters allocated to them and might be discrete or continuous. The variables are real numbers that can take on any value in a continuous range or have defined values.

Random variables are used to conduct random experiments in all forms of economic and financial decision-making. Statistical methods and probability distributions are used to estimate the most likely outcomes in a particular event, making decision-making easier.