Random Variables- What are they?

In statistical probability, a random or stochastic variable is something that changes continuously because of randomness and chance. When compared to other mathematical variables, a random one will not have a fixed, single value; rather, the random variable has the possibility to become different values that are each associated with a different probability.

The possible value of a random variable can represent many possible outcomes of an experiment or the possible outcomes of an experiment that has uncertain results such as imprecise measurements or incomplete information. Random variables can also represent an objectively random process or a subjective random process that results from not enough information of a quantity. The meaning of a particular probability that is assigned to a random variable is not necessarily part of the actual probability itself. Instead, it can be related to various philosophical arguments over the understanding of probability. Regardless, the mathematics is still going to be the same regardless of what interpretation is used.

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Random variables are categorized into two sets: discrete or continuous. A discrete variable is only a specified list of exact values, and a continuous variable is a value that is in an interval or collection of intervals. The mathematical function that is responsible is explaining the possible outcomes of a random variable and its probability is defined as the probability distribution. The results of a random variable of choosing their values according to the probability distribution is known as random variates.

In probability theory, there is a formal treatment of mathematical random variables. In this context, a random variable is defined on a sample space where the outputs are only numerical values.

An example of a random variable can include a personâ€™s height. In a mathematical sense, the random value is understood as a function that maps the particular person to their height. The random value is also associated with the probability distribution that allows the calculation of the particular probability that the height of a person is a non-pathological subset of different values. Examples include that the height is between 170 and 180 cm, or that the probability that the height is either less than 140 or more than 190 cm.

In this particular example, the sample space is suppressed since it is difficult to mathematically describe, and the possible values of the variables are then treated as the actual sample space. But when two different random variables are being measured on the same outcome, such as the height of a person, it is very easy to track the overall relationship if there is an acknowledgment that the height comes from the same random person.