Final answer:
A random variable X is a variable representing numerical outcomes of a random process, with its values and corresponding probabilities comprising its distribution. The mean of this distribution is symbolized by μ and represents the average value across all possible outcomes.
Step-by-step explanation:
Understanding Random Variables
In the context of probability and statistics, a random variable is a variable that can take on different values, each associated with a probability. One common way to represent a random variable is by using an uppercase letter, such as X, where the values it may take are denoted with a lowercase letter, such as x.
Defining Random Variable X
A random variable X can be defined as a function that maps outcomes of a random process to numerical values. The set of all possible values of X is known as its distribution.
Values and Distribution of X
For instance, if X represents the number of heads when flipping a coin three times, then the values of X could be 0, 1, 2, or 3, where each outcome has a certain probability.
The distribution of X then lists all possible values x along with their corresponding probabilities P(x). For example, if X corresponds to the number of heads in three coin flips, its distribution might show P(x=0) = 0.125, P(x=1) = 0.375, P(x=2) = 0.375, and P(x=3) = 0.125.
When one refers to the mean of a random variable, often represented by the Greek letter μ, this refers to the average value of the distribution of X, which is calculated as the sum of all products of the form xP(x).