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By definition, a continuous random variable takes on an infinite number of values within a given range. Unlike discrete random variables, which have specific, countable outcomes, continuous random variables can assume any value in an interval and are associated with probabilities described by a probability density function (PDF). In our example, the variable \(X\) represents the time a book is checked out at a library, ranging between 0 and 2 hours.
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A continuous random variable's PDF gives us a way to determine probabilities over intervals rather than at specific values. For \(X\), the PDF is defined as \(f(x) = 0.5x\), indicating how likely different checkout times are within the range \(0 \leq x \leq 2\). Understanding the behavior of \(X\) helps in solving problems like calculating specific probabilities for various intervals of time.

Integral calculus is a powerful mathematical tool used to calculate areas under curves, which in probability gives us the total probability between two points on a continuous scale. When dealing with a continuous random variable, we use integration to find the probability of the variable falling within a specific interval. In our example, to find \(P(X \leq 1)\), we compute the integral of the function \(f(x) = 0.5x\) from 0 to 1.
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The integral \(\int 0.5x\, dx\) gives us the accumulated probability, which represents the area under the curve of the density function up to that point. This is evaluated to solve each part of the problem:

• For \(P(X \leq 1)\), we integrate from 0 to 1.
• For \(P(0.5 \leq X \leq 1.5)\), we integrate from 0.5 to 1.5.
• For \(P(1.5 < X)\), we integrate from 1.5 to 2.

Each of these integrations tells us the probability of \(X\) being within the respective range, demonstrating the usefulness of integral calculus in probability.

Probability calculation for continuous random variables involves determining the likelihood that the variable falls within specified limits. Unlike discrete probabilities, probability calculations for continuous variables use integration due to the infinite range of values. For the function \(f(x) = 0.5x\), integration over a particular interval gives the probability for that interval.
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We compute the integral as follows for each probability:

• \(P(X \leq 1)\): \(\int_{0}^{1} 0.5x \, dx = 0.25\), indicating a 25% chance of checkout time being 1 hour or less.
• \(P(0.5 \leq X \leq 1.5)\): \(\int_{0.5}^{1.5} 0.5x \, dx = 0.5\), meaning a 50% probability that time falls between 0.5 and 1.5 hours.
• \(P(1.5 < X)\): \(\int_{1.5}^{2} 0.5x \, dx = 0.4375\), showing about a 44% chance of the time being more than 1.5 hours.

This illustrates how we use probability calculations in practical scenarios, transforming theoretical math into real-world context by employing calculus techniques to continuously bridging uncertainty into measurable outcomes.