Random variable can be defined as a variable that does not have a known value. It may also be defined as a function that provides values to each of the results of a particular experiment. Most of the time, letters are used to represent random variables. There are two main categories of random variables discrete and continuous. Discrete random variables are those variables that have definite values whereas continuous random variables are those that can have any values within a continuous range.
Measure theoretic definition
The measure theory is used to derive the most appropriate definition of a random variable. Sets of numbers together with functions that align these sets to probabilities are used in order to define continuous random variables. Due to the different complexities that may crop up if such sets are inadequately constrained, there may be a need to apply a sigma-algebra to constrain the sets on which probabilities can be defined. The sigma-algebra used in this scenario, known as the Borel ?-algebra, permits probabilities to be defined on any sets that can be obtained either from constant intervals of numbers or by a finite or infinite number of unions or intersections of these intervals.
Real value random variables: In scenarios where real numbers are used as the observation space, the resultant values of the function applied to the sets can be termed as real value random variables.
Distribution functions of random variables: Keeping a track of all the probabilities of output ranges of a real-valued random variable produces its probability distribution. The probability distribution does not consider the specific probability space used to define the real valued random variable, but rather takes into account the probabilities of various values of the random variable.
Functions of random variables
Random variables serve a variety of different functions. Lets take a look at a few of them.
If a real Borel measurable function is applied to the results of a real-valued random variable, a new random variable can be defined and a cumulative distribution for it can be obtained.
If the Borel measurable function is invertible, then the above distribution can be extended further.
The invertible function can also be applied to find the connection between the probability density functions by differentiating both sides with respect to the new random variable.
Equivalence of random variables
Two random variables can be considered to be as equivalent in three different ways. The variables can either be completely equal, almost surely equal or they may be equal in their distribution.
Random variables are frequently used in both mathematics as well as statistics in order to perform a variety of different functions due to their ability to provide values for the results of an experiment.