91Ó°ÊÓ

The mean, often referred to as the expected value, of a random variable is a fundamental concept in probability theory. It provides a measure to the "center" of the variable's distribution. For discrete random variables, the mean is computed using the formula \( \text{Mean} = \frac{1}{n} \sum_{i=1}^{n} x_i \), where:

• \( n \) represents the total number of different values the random variable can take.
• \( x_i \) are the individual values of the random variable.

In practical terms, the mean gives us the average value we can expect from our variable if we were to observe it repeatedly. This is especially useful to summarize data sets and predict outcomes. When each value in the set is equally likely, as seen in many textbook problems, calculating the mean is straightforward by simply summing the values and dividing by the number of values.

A discrete random variable is one that can take on a countable number of possible values. These are typically whole numbers or integers, which makes them unique when compared to continuous random variables. In the context of this exercise, the random variable \( X \) has values \([0, 1, 2, 3, x]\). Some key characteristics of discrete random variables include:

• They often arise from counting processes, such as the number of heads in a series of coin tosses.
• Probabilities are assigned to each possible value differently than with continuous variables.
• The probability distribution of a discrete random variable is often illustrated using a probability mass function, which lists the probabilities associated with each of its values.

Understanding these variables is crucial as they are commonly encountered in various fields such as statistics, computer science, and engineering.

In mathematical problems, solving an equation often involves finding unknown values that satisfy given conditions. In our exercise, the task was to find the unknown \( x \) given that the mean was 6.The solution relied on setting up an equation where\[\frac{0 + 1 + 2 + 3 + x}{5} = 6\]and then solving for \( x \). The following steps highlight the solving method:

• Simplify the equation by first calculating the sum of the known values: \(0 + 1 + 2 + 3 = 6\).
• Substitute back into the equation to obtain \(\frac{6 + x}{5} = 6\).
• Eliminate the fraction by multiplying both sides by 5, thus simplifying the equation to \(6 + x = 30\).
• Finally, solve for \(x\) by subtracting 6 from both sides, resulting in \(x = 24\).

This straightforward approach ensures you can systematically find a solution to many algebraic problems, a crucial skill in both education and practical applications.