📂Mathematical Statistics

Random Sampling in Mathematical Statistics

Definitions ¹

1. The actual outcome of a random variable $X$ is called its realization and is usually represented by the lowercase letter $x$.
2. A set of random variables from the same probability distribution as $X$, with a sample size of $n$, is called a sample, represented as follows: $$ X_{1} , X_{2} , \cdots , X_{n} $$
3. If the random variable $X_{1} , \cdots , X_{n}$ is iid, then a set with size $n$ is called a random sample.

Explanation

With these definitions, mathematical statistics establishes contact with actual statistical analysis. Handling the realization of random samples pertains to statistical analysis, and mathematical statistics serves as a significant beacon for how to manage those data. While the data of interest, the conclusions sought, and the methods employed may differ, mathematical statistics fundamentally supports the theoretical base necessary for their analysis.

In fact, outside of mathematical statistics textbooks, the term realization is rarely used, and typically, there are other terms used to refer directly to these realizations, such as values, data, or observations. However, the convention of using uppercase for random variables and lowercase for data is followed in almost all statistics textbooks.

See Also

Definition of Data in Introductory Statistics

At the undergraduate freshman and sophomore level, introductory statistics courses define data as the collection of results actually measured from experimental units or trials.