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A continuous random variable, often abbreviated as rv, is a variable that can take an infinite number of different values within a given range. This range is typically over the set of real numbers, meaning the variable can take any value within a specified interval. For example, if we are looking at temperature over a day, it can vary continuously throughout the time. When dealing with continuous random variables, we often define their behavior using a Cumulative Distribution Function (CDF). The CDF gives the probability that the random variable will take a value less than or equal to a particular number. In our exercise, we have a random variable, denoted as \(X\), with a specified CDF valid for different ranges. Understanding the behavior of \(X\) includes recognizing when the CDF assumes different values like zero or one, depending on the intervals.

The Probability Density Function (PDF) is crucial when working with continuous random variables. It tells us how the probability is distributed across different values of the variable. Essentially, the PDF describes the likelihood of the random variable taking on a specific value.In the context of our exercise, the PDF is derived from the given CDF. To find the PDF, we differentiate the CDF with respect to \(x\). This differentiation gives us a function describing the probability distribution over the continuous range. For our particular CDF, the PDF formula is determined by calculating the derivative of \(F(x)\) for \(0 < x \leq 4\), which results in:\[ f(x) = \frac{1}{4}\left[1+\ln\left(\frac{4}{x}\right) - \frac{1}{x}\right] \]This function gives us a detailed view of how probable each value within this interval is. It is important to remember that while the PDF provides insight into how data points are spread, its value is not a direct probability.

Performing probability calculations for continuous random variables often involves working with the CDF and PDF. In the step-by-step solution, we calculated the probability that \(X\) is less than or equal to a specific value and also within a range.To calculate \(P(X \leq a)\), we simply evaluate the CDF at \(a\). For instance, \(P(X \leq 1)\) is calculated using \(F(1)\), which incorporates the CDF formula given for its range. This gives us a direct probability.For range probabilities such as \(P(a \leq X \leq b)\), the computation is a matter of subtracting the CDF values: \(F(b) - F(a)\). This gives the area under the PDF curve between \(a\) and \(b\), representing that probability. For example, we calculated \(P(1 \leq X \leq 3)\) using \(F(3) - F(1)\).These methods of using the CDF and PDF for probability calculations are indispensable tools in understanding and making predictions about continuous random variables.