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A probability density function (PDF) is a fundamental concept when dealing with continuous random variables. It describes how the values of a random variable are distributed. In simpler terms, the PDF helps us understand the likelihood of different outcomes in a continuous sample space.

The function itself, which we denote as \(f(x)\), assigns a probability to intervals of values, rather than individual points. This is because, in continuous distributions, the probability of the variable taking an exact value is zero.

One important property of a PDF is that the total area under the curve of this function is equal to 1. This represents the certainty that some value within the existing range of possibilities will occur. In our exercise, this verification is a crucial step in ensuring the PDF is valid. It is achieved by integrating the function over its defined range, checking that the result is indeed 1.

A piecewise function is a type of mathematical function defined by multiple sub-equations or "pieces," each applying to a certain interval within the function's domain. This is especially useful when dealing with probability functions that may vary in behavior within different ranges of a random variable.

In our given problem, the probability density function \(f(x)\) is piecewise, which means it takes different forms depending on the value of \(x\). Specifically, it is defined as a linear function \(0.075x + 0.2\) within the interval from 3 to 5, and equals zero outside of this interval.

Piecewise functions are common in continuous probability scenarios because they allow the PDF to effectively model scenarios where the distribution of values can change significantly over different ranges.

The concept of the area under the curve (AUC) is vital when working with probability density functions. It represents the total probability of a random variable falling within a certain range.

In terms of calculus, AUC involves integrating the PDF over the desired interval of the random variable. This mathematical process allows us to calculate the probability for continuous distributions. For instance, in our exercise, to verify the valid PDF and calculate probabilities like \(P(X \leq 4)\), we integrate the function over the specified range.

By evaluating these areas, we can effectively determine how likely an event is to occur within certain confines, reflecting the nuanced variations of probability in continuous spaces.

Calculating probabilities in the context of continuous random variables often involves integrating the probability density function over a given interval. This calculated area corresponds to the likelihood that the random variable falls within that interval.

For the exercise in question, we perform these calculations by setting the lower and upper bounds for integration according to the desired interval. For example:

• To compute \(P(X \leq 4)\), we integrate from 3 to 4.
• For \(P(3.5 \leq X \leq 4.5)\), we set boundaries at 3.5 and 4.5.
• Finally, to find \(P(4.5 < X)\), integration is done from 4.5 to 5.

Continuity of the random variable allows us to equate \(P(X \leq a)\) with \(P(X < a)\), simplifying calculations unless specifically stated otherwise. This methodology ensures all probabilities are accurately determined, respecting the continuous nature of the variable.