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A probability density function (PDF) helps describe the likelihood of different outcomes for a continuous random variable. Unlike a probability mass function, which is used for discrete variables, the PDF integrates over an interval to give the probability of the random variable falling within that interval.

For our given problem, the random variable \(X\) follows a uniform distribution over the interval \([-1, 1]\). Uniform distributions imply that each outcome in the interval is equally likely. The PDF for a uniform distribution is constant over its range, computed by the formula:

• \(f(x) = \frac{1}{b-a}\) for \(x\in [a, b]\)

In this case, since \(X\) is uniformly distributed over \([-1, 1]\), the PDF is:

• \(f(x) = \frac{1}{2}\) for \(x \in [-1, 1]\)
• \(f(x) = 0\) otherwise

With this PDF, we can compute probabilities for various outcomes within the interval using integration. This step is crucial for understanding the transformation of \(X\) to \(Y\).

The cumulative distribution function (CDF) of a random variable describes the probability that the variable will take a value less than or equal to a specific value. It's a function that increases monotonically from 0 to 1 as you traverse the range of the variable.

In our exercise, we calculate the CDF for the transformed variable \(Y\). It's constructed as follows:

• For values \(y < -0.5\), the CDF \(F_Y(y) = 0\) because there's no chance for \(Y\) to be less than \(-0.5\).
• For \(-0.5 \leq y < 0.5\), the CDF \(F_Y(y) = \frac{1}{4} + \frac{1}{2}(y + 0.5)\). This part accounts for the transformation of \(X\) when it directly equals \(Y\) between the values \(-0.5\) to \(0.5\).
• At \(y = 0.5\), \(F_Y(0.5) = \frac{3}{4}\), reflecting the cumulative probability of all realizations of \(X\) moving to \(Y\).
• For \(y > 0.5\), the CDF \(F_Y(y) = 1\), indicating that all the possible values \(Y\) could assume have been accounted for.

Graphing this CDF visually represents the cumulative probability of \(Y\) as various values, demonstrating both smooth increases and jumps, notably the jump at \(y = 0.5\).

Transforming random variables involves creating a new variable based on another variable according to a specific rule or function. In this problem, the random variable \(Y\) is derived from \(X\) using a 'hard limiter' function.

The transformation rules defined in the problem state that:

• \(Y = X\) if \(|X| \leq 0.5\)
• \(Y = 0.5\) if \(X > 0.5\)
• \(Y = -0.5\) if \(X < -0.5\)

This alters the distribution of \(X\) into a new distribution for \(Y\). Such transformations are common in applications where signals need to be modified or 'clipped' to remain within a specific range, affecting the subsequent probability distributions.

Analyzing the transformed variable \(Y\) involves determining its probability characteristics, such as the probability of being in certain intervals (for example, \(P(Y = 0.5) = \frac{1}{4}\)). A good understanding of these transformations aids in predicting system behaviors, such as in electronic and signal processing applications.