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Random Variables and Random Processes

In rough terms, a random or stochastic variable can be termed as the variable whose values are the result of some type of random processes. It can be said to be a function of probability space and is measurable, which makes it typical to real numbers. Random variable can also be said to be a numerical description of the outcome of a certain experiment. Random variables can either be discrete or continuous. The random variable which assumes either a finite set or finite sequence of values is called discrete, whereas the continuous variables are those that may assume any numerical value in a single or a collection of intervals.

The discrete random variable maps the outcome of various values of a countable set for example the integers and each value is in the range of having probability either greater than or equal to zero. In the case of continuous random variable, the mapping is of the outcomes to values of an uncountable set. For a continuous random variable, the probability of any specific value is zero but the probability of some finite set of values can be positive.

The possible values of a random variable represent the possible values of the outcomes of the experiments which are yet to be performed. The realizations of random variables are termed as random variate also. The random variable can also be thought of as a quantity which has unfixed value.

The real valued random variables are used in science for making predictions on the basis of data obtained from scientific experiments. They were initially developed for the analysis of the games of chance and for other stochastic events.

If the probability space is and the measurable space is can be said to be a function of measurable. The meaning of the latter is that for any and every subset BEℰ, the prime age is By this definition we can measure any element B in the target space, by looking at the preimage, which is measurable by assumption. When the topological space is E, then the most common choice of the σ-algebra is to make it equal to the Boral algebra (E). The collection of all open sets in E, make up the σ-algebra. The E, valued random variable is called the E valued random variable. When the space e is the real line, it is called simply as the random variable. Can be said to be a real valued random variable when For the process of generalization of assigning a given value to any random variable, a cumulative distribution function is associated with the random variable. In case the cumulative distributive function is a heavy side step function. Then the probability is 1 and the variable takes the value at the jump. Generally, CDF specifies the probability which the variable take son particular values. If we define a random variable as on a given probability space then the question is "what is the likeliness of the value of X being bigger than 2?” The probability is same for the event which can be written is short as and can be easily obtained as Thus by recording all these probabilities for various output ranges for a real valued random variable X can yield the probability distribution of X. Many a times, the probability distribution of these random variables is characterized by some small number of parameters. These parameters have a practical interpretation also. The average value when captured by the mathematical concept of the expected value of any random variable is denoted by E(X) and is called as the first moment.

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