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A probability density function (pdf) is a function that describes the likelihood for a random variable to take on a particular value. For continuous random variables, the pdf, denoted as \( f(x) \), represents the density of probability rather than direct probabilities themselves. The value of a pdf at any specific point conveys how likely a value close to that point is to occur. The area under the curve of a pdf over an interval gives the probability that the random variable falls within that interval.

When we have a uniform distribution, such as the one for random variable \( X \) distributed uniformly on \([-1, 1]\), the probability density function is constantly valued across its support. In this case, the pdf is \( f_X(x) = \frac{1}{2} \) for \(-1 \leq x \leq 1\). This flat line indicates equal probability across all values in the interval. To find the pdf for another variable, such as \( Y = |X| \), we transform the variable and calculate its new distribution from the original pdf.

The cumulative distribution function (CDF), denoted as \( F(x) \), gives the probability that a random variable \( X \) is less than or equal to a certain value \( x \). It is obtained by integrating the pdf over a range from the lowest possible value up to \( x \). An important characteristic of CDFs is that they are non-decreasing and always range from 0 to 1.

In this exercise, we deal with a random variable \( Y = |X| \). The CDF of \( Y \), \( F_Y(y) \), is found by calculating the probability \( P(|X| \leq y) \). For \( X \) uniformly distributed on \([-1,1]\), the expression becomes \( F_Y(y) = \int_{-y}^{y} \frac{1}{2} \, dx \), which simplifies to \( y \) for \( 0 \leq y \leq 1 \). This piecewise function indicates that for \( y < 0 \), \( F_Y(y) = 0 \), and for \( y > 1 \), \( F_Y(y) = 1 \), as probabilities beyond these ranges are either improbable or certain.

In statistics, transformation of random variables involves creating a new variable by applying a function to an existing one. This method allows us to derive the distribution of a new random variable from a known distribution. In this exercise, we derived \( Y = |X| \) from \( X \), which was uniformly distributed on \([-1, 1]\).

To determine the new distribution, we first found the CDF of \( Y \), \( F_Y(y) \), by setting up the probability statement \( P(|X| \leq y) \) and integrating the known pdf of \( X \) over this range. After obtaining the CDF, differentiating it gave us the pdf of \( Y \). This transformation technique is particularly useful when dealing with non-linear functions or absolute values, like \( |X| \), because it enables us to analyze properties and calculate probabilities of new distributions formed from transformations.