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In probability theory, a uniform distribution describes a situation where every outcome in a given interval has an equal chance of occurring. Specifically, when we say a random variable \(X\) follows a uniform distribution on the interval \([-1, 1]\), we mean that any value within this interval is equally likely to be the outcome.

To imagine this, consider a perfectly balanced die where each side represents a possible outcome within the interval. The length of the interval is 2 units, but because the distribution is uniform, each unit has a probability of \(\frac{1}{2}\). For problems involving uniform distributions, probabilities are often calculated by determining the length of sub-intervals of interest, divided by the total length of the overall interval.

This equal distribution of probability simplifies many calculations, as seen in the exercise where the random variable \(X\) is initially defined.

The cumulative distribution function (CDF) is a fundamental concept in probability that provides the probability that a random variable \(Y\) will take a value less than or equal to a specific point. Essentially, it gives us the "accumulated" probability up to that point.

In the case of our uniform distribution, the CDF of the random variable \(Y\) was constructed to represent the different transformations applied by the hard limiter.

• For values of \(y < -0.5\), the probability \(P(Y \leq y)\) is 0, because \(Y\) cannot be less than \(-0.5\).
• For values within the interval of \(-0.5\) to \(0.5\), the probability \(P(Y \leq y)\) gradually increases from 0 to 0.75.
• Exactly at \(y = 0.5\), the probability reaches 0.75, incorporating results up to and including \(Y = 0.5\).
• For values \(y > 0.5\), the probability \(P(Y \leq y)\) becomes 1, meaning all possible outcomes for \(Y\) have been covered.

This piecewise function helps us visualize the probability variations across different output values of \(Y\).

Calculating probabilities within the context of uniform distributions involves understanding the proportion of the interval relevant to the scenario. For example, in determining \(P(Y = 0.5)\), it's essential to look at where \(Y = 0.5\) according to the conditions defined by the hard limiter.

The random variable \(Y\) reaches the value 0.5 when \(X > 0.5\). Given the uniform distribution over \([-1, 1]\), the interval \([0.5, 1]\) represents this scenario. Since this interval has a length of 0.5 relative to the total interval of 2 units, \(P(Y = 0.5) = \frac{0.5}{2} = 0.5\).

In summary, identify the relevant sub-interval for the event, determine the length of this interval, and divide by the total length of the uniform distribution to find the probability.

Random variables are a core concept in probability that map outcomes of random processes to numerical values. They allow us to mathematically study and model randomness.

In this exercise, the random variable \(X\) represents the voltage at the output of a microphone, with defined possible values between \(-1\) and \(1\). As a transformation, the hard limiter introduces another random variable \(Y\), which maps the outputs of \(X\) to three specific states based on a set of predefined conditions.

• If \(|X| \leq 0.5\), then \(Y = X\).
• If \(X > 0.5\), \(Y = 0.5\).
• If \(X < -0.5\), \(Y = -0.5\).

By establishing these conditions, \(Y\) behaves as a discrete random variable, albeit as a function of the original continuous random variable \(X\). Understanding transformations between random variables is critical in situations like signal processing and other applied sciences, where it is common to map outputs through various filters and conditions.