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The probability mass function, or pmf, is a fundamental concept in probability theory, especially when dealing with discrete random variables. A pmf gives the probability that a discrete random variable is equal to a particular value. In simple terms, it tells us how the probabilities are distributed over the values of the random variable.

To apply a pmf, you need a few key components:

• Each possible outcome of the variable.
• The probability associated with each outcome.
• The sum of the probabilities must equal 1, as the outcomes cover the entire sample space.

In the exercise, the pmf is given as \( p(x) = \frac{1}{5} \) for \( x = 1, 2, 3, 4, 5 \) and zero elsewhere. This indicates each of these five outcomes has an equal probability of occurring.

When working with pmfs, make sure each probability assigned logically represents the likelihood of occurrence and the sum of all probabilities equals 1.

A discrete random variable, as the name suggests, can take on a finite or countably infinite number of distinct values. These values are separate and unconnected, which means the variable will jump from one value to another without taking intermediate values, unlike continuous random variables.

Think of a discrete random variable as something you can count. For example:

• The number of students in a classroom.
• The result of rolling a six-sided die.
• The number of heads when flipping a coin three times.

In our exercise, the random variable \( X \) is discrete because it only takes one of the five distinct values: 1, 2, 3, 4, or 5. There are no values between these numbers that \( X \) can take.

Understanding the concept of discrete random variables helps us apply tools like the probability mass function effectively. It is important to recognize whether a random variable is discrete or continuous to choose the appropriate mathematical approach.

The expected value of a random variable is a crucial concept in probability and statistics. It represents the average or mean value that one would expect from a probability distribution in the long run. For discrete random variables, the expected value is computed using the formula:\[ E(X) = \sum (x_i * p(x_i)) \]where \( x_i \) are the possible values of the random variable and \( p(x_i) \) is the probability of each value.

In the exercise, the expected value of the discrete random variable \( X \) is calculated using the provided pmf. The process involves multiplying each value \( x \) by its probability \( p(x) \) and then adding these products together. For \( X \), this computed expected value is 3, indicating that on average, you'd expect the outcome of \( X \) to be around 3.

For higher power terms like \( X^2 \), the concept stays consistent. You compute the expected value using the function of the random variable, \( E(X^2) \), as done in the exercise. These computations help in understanding how spread out a distribution is and are widely used in theoretical and real-world applications.