Probability theory is a branch of mathematics that deals with the description and analysis of random events. The key building blocks of this framework are as follows:

A random variable is a quantity whose value is random or unpredictable, and to which we can assign a probability distribution function.

The probability distribution function determines the set of possible values that can be assigned to the random variable, along with their likelihood. The total of all possible outcomes’ likelihood must by definition equal one (i.e., one of the possible outcomes must happen, and its value will be assigned to the random variable).

Let’s use a fair gaming die as an example. The top face of the die can take on one of six possible outcomes (i.e., one, two, three, four, five, six). The probability distribution function is uniform (i.e., there is an equal one-in-six chance of any value between one and six coming up). When you sum up all of the possible probabilities, they add up to exactly one.

Stochastic Processes

There are two kinds of processes considered in probability theory: deterministic and stochastic.

A deterministic process will go along a set path depending on its starting conditions. In other words, if you know where it starts, you can exactly compute where it will end up at some point in the future.

A stochastic process (also called a random process) is more difficult to understand. You cannot tell exactly where it will end up, but you know (based on its probability distribution function) that certain outcomes are more likely. In the simplest case, a stochastic process can be described as a sequence of samples from random variables. If these samples can be associated with particular points in time, it is a time series (a series of data points that were measured at successive times). [Read the rest of the article here]

 

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