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Probability calculation often involves determining the likelihood of certain events occurring based on the known data and statistical models. In the context of cumulative distribution functions (CDF), calculating probabilities requires understanding the relationship between the CDF and the probabilities of events involving a random variable. For a random variable \(X\), the CDF \(F(x)\) gives the probability that \(X\) takes a value less than or equal to \(x\). Hence, the probability \(P(X < x)\) can be directly derived from \(F(x)\) when \(x\) is within the domain of the function.
To calculate the probability of a range, such as \(P(a < X < b)\), we find \(F(b)\) and \(F(a)\), and then subtract the two: \(P(a < X < b) = F(b) - F(a)\).
This method allows for straightforward computation using the CDF, as seen in the solutions for the problems like finding \(P(X < 1.8)\) and \(P(-1 < X < 1)\). This illustrates how understanding the role of the CDF facilitates the computation of probabilities over both individual values and intervals.

Random variables are a fundamental concept in probability and statistics. They represent numerical outcomes of a random phenomenon. When dealing with random variables, it's crucial to understand their distribution, which describes how probabilities are assigned to each possible value of the random variable.
In the problem provided, the random variable \(X\) has a distribution defined by a piecewise CDF. This distribution tells us how \(X\) behaves over different intervals. A cumulative distribution function, like the one given in the exercise, sums probabilities up to a certain point, helping us understand which values \(X\) might realistically take.
Random variables can be discrete or continuous. In this case, the random variable \(X\) is continuous, as indicated by the continuous range of values it can take within specified intervals. Understanding the behavior and characteristics of random variables is essential for accurate probability calculations and statistical analysis.

Piecewise functions are functions defined by multiple sub-functions, each applying to a certain interval of the domain. In the context of a cumulative distribution function, a piecewise function allows for modeling of complex systems where the behavior varies over different ranges.
In the original exercise, the piecewise CDF for the random variable \(X\) is described as follows: \(F(x) = 0\) for \(x < -2\), \(F(x) = 0.25x + 0.5\) for \(-2 \leq x < 2\), and \(F(x) = 1\) for \(x \geq 2\). This allows the probability distribution to capture different scenarios that \(X\) may encounter.
When analyzing such piecewise functions, it's important to determine which part of the function applies to a given evaluation point. This requires understanding both the mathematical structure of the function and the contextual meaning of each section. Successfully using piecewise functions involves calculating probabilities or expected values by leveraging the defined sub-functions correctly, ensuring precise solutions, like determining \(P(X < 1.8)\), which falls under the interval \(-2 \leq x < 2\).
By dissecting each segment of the piecewise function, one can fully grasp its overall impact on the probability calculations.