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A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. In probability theory, we often deal with random variables to model real-world scenarios. They can be either discrete or continuous.

- **Discrete Random Variables** take on a countable number of distinct values, like the roll of a die.- **Continuous Random Variables** can take on any value within a given range, such as a person's height.
In our problem, the side of the square, denoted by the random variable \(X\), is continuous since it can take any value between 0 and 4. The area \(Y\), which depends on \(X\), is also a random variable. Understanding how these random variables interact and relate to one another is fundamental in calculating probability distributions.

The Cumulative Distribution Function (CDF) of a random variable describes the probability that the variable takes a value less than or equal to a given number. For a continuous random variable like \(Y\), the CDF is crucial for understanding the behavior of \(Y\) over its range.

In this problem, the CDF of \(Y\), denoted as \(F_Y(y)\), is derived from the CDF of \(X\) due to the relationship \(Y = X^2\). We express \(F_Y(y)\) as \(P(Y \leq y) = P(X^2 \leq y)\), which equals \(P(X \leq \sqrt{y})\). Calculating this results in:

• \(F_Y(y) = \int_{0}^{\sqrt{y}} \frac{x}{8} \, dx = \frac{y}{16}\) for \(0 < y < 16\).

The CDF is essential because it provides a comprehensive description of the distribution of \(Y\), helping us understand all possible outcomes and their likelihoods.

Integration plays a vital role in probability, especially when working with continuous random variables. It is used to derive various functions that are fundamental in probability theory, such as CDFs and PDFs.

Here, integration was used to compute \(F_Y(y)\), the CDF of \(Y\), from \(f_X(x)\), the PDF of \(X\). This step is critical because:

• It translates the PDF, which tells us only the density, into a CDF, which shows the cumulative probability up to a point \(y\).
• To get from the CDF back to the PDF of \(Y\), denoted \(f_Y(y)\), you differentiate the CDF: \(f_Y(y) = \frac{1}{16}\) for \(0 < y < 16\).

Moreover, integration confirms the PDF is valid by ensuring the total area under the curve for \(f_Y(y)\) equals 1. Thus, the role of integration extends across evaluating probabilities, validating distributions, and transitioning between function types.