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A discrete random variable is a fundamental concept in probability and statistics that represents a variable with a countable number of distinct outcomes. Imagine it like rolling a die; the outcome can only be one of the six integers (1, 2, 3, 4, 5, or 6). In the context of the given exercise, the random variable \(X\) is limited to the values 0, 1, 2, 3, and 4. Each of these values has a certain probability assigned by the function \(f(x) = \frac{2x+1}{25}\).

Discrete random variables are often analyzed using probability mass functions (PMFs). A PMF provides the probability that a discrete random variable is exactly equal to some value. This characteristic makes PMFs extremely useful in various applications such as quality control, risk assessment, and in optimizing business processes. Always remember, a discrete random variable deals with events that occur in distinct steps.

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The core idea here is to model uncertainty in a mathematically logical way. Events in probability theory are unpredictable in the short term, but fairly predictable in the long term when patterns emerge.

In our exercise, to understand how likely each outcome is, we use probability mass functions. For example, calculating \(P(X=4)\) means determining how likely it is that \(X\) equals 4. Probability theory not only helps us define these specific probabilities, but it also assists in understanding larger concepts like cumulative probabilities, independence, and expected values.

By verifying that the total sum of probabilities equals 1, we ensure the setup adheres to the laws of probability, indicating no outcome is overlooked and that probabilities are correctly distributed across all possibilities.

Probability calculation is the practical application of probability theory to determine the likelihood of different events. In the context of our problem, after confirming that the function \(f(x)\) behaves as a valid probability mass function, we use it to calculate specific probabilities. Such calculations can answer questions about distinct outcomes or ranges of values.

For instance, calculating \(P(X \leq 1)\) requires sum probabilities of \(f(0)\) and \(f(1)\). This is an example of determining cumulative probabilities, where you find the total probability for a range of outcomes. Similarly, the probability \(P(2 \leq X < 4)\) is calculated by adding \(f(2)\) and \(f(3)\), representing probabilities within a specified range.

These calculations help provide clear, quantitative insights into the likelihood of various scenarios, making them crucial for decision-making processes wherever uncertainty is present.