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Brownian Motion
Brownian motion is a concept in physics that is used to explain why some quantity is consistently undergoing small but random changes. Robert Brown was the first person to study these changes in 1827. The classical Brownian motion explains the random motion of the small particles in liquid or gas. Mr. Brown saw a rapid oscillatory movement of microscopic particles in the pollen grains suspended in water when he was investigating the process of fertilization in Clarkia Pulchella.
Probability Spaces
A probability space is used to model random events. It is subdivided into three parts:
Sample space
It is the group of all possible results. Suppose we toss a coin twice, the sample space will be (HH, TT, TH, and HT). The Greek letter omega is sometimes used to denote sample space.
Event space
Event space is the set that has all the events. Event space can be zero or any number.
Probability function
A probability function assigns a probability to the phenomenon. For example, if we toss a coin, the chance that we will get a "head" is 50%.
Random Variables
A random variable numerically describes the results of a statistical experiment. A variable is said to be discrete when it only assumes a finite or infinite series of values. On the other hand, a continuous variable takes any real number line value in some interval. For example, we can describe the number of vehicles sold by a motor dealership in a day as a random discrete variable, while the random weight in kilograms or pounds of a person at the hospital would be a random continuous variable.