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A probability mass function (PMF) is an essential concept when dealing with discrete random variables. It tells us the probability that a discrete random variable is exactly equal to some value. In more straightforward terms, think of PMF as translating the likelihood of each possible outcome of a mathematical experiment into numbers that add up to one.
The PMF of our variable, say \(X\), can be formalized as \(P(X = x)\), where \(x\) represents the possible values our random variable can take.
In the provided exercise, we have observed such a translation through different intervals. For instance, \(P(X = 1)\) is 0.30, implying that the probability of a policyholder making payments every month is 0.30.
Similarly, other PMF values are calculated based on the changes between cumulative distribution values, showing the jumps from one interval to the next.

The cumulative distribution function (CDF) is pivotal in understanding how probabilities accumulate over a range of values. For a random variable \(X\), the CDF, denoted as \(F(x)\), represents the probability that \(X\) will take a value less than or equal to \(x\).
This helps us when we need to calculate probabilities over intervals rather than single points. In the given exercise, the CDF is defined piecewise, showing how probabilities build up as \(X\) reaches different thresholds.
To compute probabilities between two points, we rely on the difference between their CDF values. For example, \(P(3 \leq X \leq 6)\) was found by subtracting the cumulative probability at X equals just less than 3 from that at 6, yielding a 0.30 cumulative probability for that segment.

A random variable is a fundamental concept in probability and statistics, representing outcomes of a mathematical experiment numerically. It serves as a function that assigns numerical values to each event in a sample space, allowing us to analyze more intricate probability scenarios.
In the context of the exercise, \(X\) is our random variable, reflecting the intervals between payment options for insurance policyholders. Different intervals indicate varied payment structures directly linked to the probabilities defined by the PMF and CDF.
By understanding \(X\) as a function that can take on specific values based on a given distribution, students can correlate the probabilities of these payment intervals in real-world scenarios, aiding their grasp of randomness and probability calculations in practical applications.