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A probability density function (pdf) is a powerful tool in understanding continuous random variables. It describes the likelihood of a random variable taking on a particular value. To grasp it, imagine spreading a finite amount of probability (which totals to 1) along a line representing all possible outcomes.

In the case of a continuous random variable, the pdf gives us a curve on a graph, where the total area under the curve is equal to 1. This graphical representation is given by the function denoted as \( f(x) \). Keep in mind that for any specific value of \( x \), the pdf doesn't give us the probability itself—since for a continuous variable, the probability of it taking an exact value is zero. Instead, it helps us determine the probability of the variable falling within an interval by integrating the pdf over that interval.

For instance, to find the probability that a random variable \( X \) falls between \( a \) and \( b \), one would calculate the integral of \( f(x) \) from \( a \) to \( b \), symbolically written as \( \int_{a}^{b} f(x) dx \). Importantly, for the pdf to be valid, it must be non-negative for all \( x \), and the area under the entire curve must equal to 1.

When we discuss a symmetric probability distribution, we are referring to the characteristic of a pdf where it mirrors itself across a central value, denoted as \( a \) in our exercise. The symmetry in the distribution means that the probability of observing a value at an equal distance from the center point \( a \) is the same on both sides.

This can be mathematically represented for a continuous random variable \( X \) with a symmetric pdf \( f(x) \) about \( a \) as \( f(x-a)=f(-(x-a)) \). In simpler terms, if you were to fold the graph of the pdf at the value \( a \), both halves would match perfectly.

Symmetry in probability distributions simplifies many calculations and can provide insights into the behavior of the random variable. For example, if a pdf is symmetric about its mean, then the mean and median of the distribution are the same, and the skewness of the distribution is zero, indicating no tail is longer or fatter than the other.

Transformation of random variables is a technique to create a new random variable from an existing one by applying a mathematical function to it. Think of it like changing the scale of a ruler – the distances between points stay the same, but you're reading them in a different unit.

In the exercise, we perform transformations by subtracting a value \( a \) from \( X \), to get a new variable \( Y \), and by negating \( X - a \) to create variable \( Z \). Mathematically, this is defined as \( Y = X - a \) and \( Z = -(X - a) \). These transformations are useful because they can simplify complicated distributions or allow us to use standard results and techniques.

Applying a transformation changes its pdf - but how it changes depends on the transformation. For linear transformations like these, the new pdf can be found by replacing \( X \) with the transformation expression. For example, the pdf of \( Y \) is denoted as \( g_Y(y) = f(y+a) \), showing that to find \( g_Y(y) \), we input \( y+a \) into the original pdf function \( f \). If the original variable had a symmetric distribution, such transformations can help prove that the transformed variables have equivalent distributions as seen in our exercise.