6.2:
Random Variables
Consider rolling a die thirty times. In each trial, the outcome can be anything from one to six. If one comes up six out of thirty times, its probability is six over thirty, and so on.
Each of these outcomes, known as random variables, has a single numerical value determined by chance. It represents all the possible outcomes of an experiment.
The lower case letter x denotes the numerical value of the random variable.
Random variables can be discrete or continuous.
Discrete random variables can be associated with a counting process, either finite or infinite. For example, a hen may lay one egg, two eggs, or more, but not 1.27 eggs.
Conversely, continuous random variables have infinitely many values that can be associated with measurements without gaps or interruptions on a continuous scale.
For example, in a day, a cow may produce anywhere between zero to twenty liters of milk, measured on a continuous scale.
6.2:
Random Variables
A random variable is a single numerical value that indicates the outcome of a procedure. The concept of random variables is fundamental to the probability theory and was introduced by a Russian mathematician, Pafnuty Chebyshev, in the mid-nineteenth century.
Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words, and x is given as a number.
For example, let X = the number of heads you get when you toss three fair coins. The sample space for tossing three fair coins is TTT; THH; HTH; HHT; HTT; THT; TTH; HHH. Then, x = 0, 1, 2, 3. X is in words, and x is a number. Notice that for this example, the x values are countable outcomes.
Random variables can be of two types: discrete random variables and continuous random variables.
A discrete random variable is a variable that has a finite quantity. In other words, a random variable is a countable number. For example, the numbers 1, 2, 3,4,5, and 6 on a die are discrete random variables.
A continuous random variable is a variable that has values from a continuous scale without gaps or interruptions. A continuous random variable is expressed as a decimal value. An example would be the height of a student – 1.83 m.
This text is adapted from Openstax, Introductory Statistics, section. 4 Introduction