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The density function of a random variable is a crucial part of probability theory. It defines how the values of a random variable, say \(Y\), are distributed across different intervals. The density function you're dealing with here, \(f(y)\), is specifically defined for \(-1 \leq y \leq 1\), and it is zero elsewhere.
This means that \(Y\) can only take values between -1 and 1, and this is where we should focus our attention. The function you have is \(f(y) = \frac{2}{\pi(1+y^{2})}\), which indicates the likelihood of \(Y\) being around any point \(y\) in the interval. Since it's a density function, when you integrate it over the entire space, it should equal to 1. Thus:

• The function maps each point \(y\) to its likelihood within the specified interval.
• For points outside \( -1 \leq y \leq 1\), the likelihood is zero, given by \(f(y) = 0\).

Understanding the density function helps us to determine other aspects like cumulative distribution function and expected value, which give us better insights into the behavior of the random variable \(Y\).

The Cumulative Distribution Function (CDF), denoted as \(F(y)\), is a function that describes the probability that your random variable \(Y\) will take a value less than or equal to \(y\).
As calculated in the solution, the CDF for the given density function is derived by integrating \(f(y)\) from the lower bound up to \(y\). This integration results in the function \(F(y) = \frac{\tan^{-1}(y)}{\pi} + \frac{1}{4}\), valid for \(-1 \leq y \leq 1\).

• The CDF informs us how probabilities accumulate as \(y\) increases from the lower to the upper bound.
• At \(y = -1\), the CDF value is 0, indicating no probability accumulated below \(y = -1\).
• At \(y = 1\), the CDF equals 1, representing the total probability of 1 when all possibilities are considered.

Exponentially growing as you move from -1 to 1, the CDF is a helpful tool for understanding which values of \(Y\) are more probable as compared to others. It climbs smoothly as \(y\) increases, reflecting how the likelihood builds up.

The expected value, also known as the mean or average value, is a key concept in probability theory. It's like the center of gravity for the random variable representing its "average" outcome across many repetitions of a scenario.
For a continuous random variable like \(Y\), the expected value is computed by integrating the product of the variable and its density function over the range of \(Y\). In simpler terms, it is the sum of each possible value weighted by its probability.
In the exercise, the expected value \(E(Y)\) of the random variable \(Y\) is computed by:

• integrating the expression \(y \cdot f(y)\) over its defined interval \([-1, 1]\).
• Given the symmetry in the density function around zero, the expected value becomes 0 without detailed calculation.

This indicates that \(Y\), on average, balances out to zero, equally likely to be positive or negative within its given interval. The expected value provides insights into the distribution's "center point" and is useful for various statistical and engineering applications.