Random Variables

- Overview

A random variable is a variable that assigns a numerical value to each outcome of an experiment. The value of a random variable changes with each trial of the experiment. 

Random variables can be either discrete or continuous: 

• Discrete: A random variable that can only take on a finite number of values or an infinite sequence of values. Examples of discrete random variables include the number of children in a family, the Friday night attendance at a cinema, the number of patients in a doctor's surgery, the number of defective light bulbs in a box of ten.
• Continuous: A random variable that can take on any value in a continuous range. For example, the height of students in a class, the amount of ice tea in a glass, the change in temperature throughout a day, and the number of hours a person works in a week all contain a range of values in an interval, thus continuous random variables.

Random variables are usually represented by capital letters near the end of the alphabet, such as X, Y, or Z. 

For example, X could be defined as the number of ice creams a customer orders. X is a random variable because it can take on several different values and is the result of a chance happening. 

The concept of "randomness" is fundamental to the field of statistics. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward.

Please refer to the following for more information:

• Wikipedia: Random Variable

- Discrete and Continuous Random Variables

A random variable is a numerical description of the outcome of a statistical experiment. A random variable that may assume only a finite number or an infinite sequence of values is said to be discrete; one that may assume any value in some interval on the real number line is said to be continuous. 

For instance, a random variable representing the number of automobiles sold at a particular dealership on one day would be discrete, while a random variable representing the weight of a person in kilograms (or pounds) would be continuous. 

The probability distribution for a random variable describes how the probabilities are distributed over the values of the random variable. For a discrete random variable, x, the probability distribution is defined by a probability mass function, denoted by f(x). This function provides the probability for each value of the random variable. 

In the development of the probability function for a discrete random variable, two conditions must be satisfied: 

• f(x) must be nonnegative for each value of the random variable.
• the sum of the probabilities for each value of the random variable must equal one.

A continuous random variable may assume any value in an interval on the real number line or in a collection of intervals. Since there is an infinite number of values in any interval, it is not meaningful to talk about the probability that the random variable will take on a specific value; instead, the probability that a continuous random variable will lie within a given interval is considered.

- The Probability Distribution for A Random Variable

The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. 

For a discrete random variable, the probability distribution is defined by a probability mass function, denoted by f(x). The probability distribution for a discrete random variable can be represented by a formula, a table, or a graph. 

The probability distribution of a discrete random variable X is a listing of each possible value x taken by X along with the probability P(x) that X takes that value in one trial of the experiment. 

The probability distribution function is also known as the cumulative distribution function (CDF). The probability distribution function gives the probability that X will take a value lesser than or equal to x. 

Here are some steps for constructing a probability distribution for a discrete random variable:

• List out all possible outcomes of the experiment.
• Count the total number of outcomes and calculate the probability of each outcome.
• Display the information in a histogram with probabilities on the vertical axis and outcomes on the horizontal axis.

The probability distribution of a discrete random variable X is a list of each possible value of  X together with the probability that  X takes that value in one trial of the experiment.

The probabilities in the probability distribution of a random variable X must satisfy the following two conditions:

• Each probability  P(x) must be between  0 and  1: 0≤P(x)≤1. 
• The sum of all the possible probabilities is  1: ∑P(x)=1.

A random variable's probability distribution describes how probabilities are distributed across the values of the random variable. The probability distribution of a discrete random variable can be represented by a formula, a table, or a graph. 

For example, if the random variable Z is the number on the top face of a die when it is rolled once, the possible values for Z will be 1, 2, 3, 4, 5, and 6. 

All random variables (discrete and continuous) have a cumulative distribution function. This function gives the probability that the random variable X is less than or equal to x, for every value x. 

Here are some examples of random variables that are normally distributed: Height, Birth weight, Reading ability, Job satisfaction, SAT scores.

[More to come ...]