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The cumulative distribution function (CDF), denoted as \( F(y) \), is a key concept in probability that provides the probability that a random variable \( Y \) is less than or equal to a specific value \( y \). In simpler terms, it gives us the cumulative probability up to a point. To find the CDF from a probability density function (PDF), we integrate the PDF over the range from the lowest value up to \( y \). This integration process accumulates the probabilities into the CDF. For our specific function, integrating \( f(y) \) from 0 to \( y \) results in \( F(y) = \frac{1}{4}y^2 \) for \( 0 \leq y \leq 2 \). This squared relationship implies that the CDF isn't a straight line, but rather a curve that steepens as \( y \) increases, reflecting increasing total probability. The CDF is 0 for \( y < 0 \) and 1 for \( y > 2 \), as the probability reaches its full value in the defined range.

Integration in probability is a fundamental tool, especially when working with continuous random variables. The idea here is to sum up probabilities over an interval to find total probabilities or to derive other related functions like the CDF. In our problem, we integrated the PDF \( f(y) = \frac{1}{2}y \) from 0 to \( y \) to find the CDF, \( F(y) = \frac{1}{4}y^2 \). This means we're accumulating the probability density into a total probability up to \( y \). When dealing with PDFs, remember that the entire area under the curve over its defined range must equal 1 to satisfy the properties of a probability density function. This explains why the integral of a PDF over its entire support gives 1. Thus, mastering basic integration techniques helps us solve various problems involving continuous random variables.

Graphing probability functions provides a visual understanding of how probabilities are distributed. For a probability density function like \( f(y) = \frac{1}{2}y \), the graph shows a line starting from the origin and rising linearly to the point (2,1). Beyond the range \( 0 \leq y \leq 2 \), the function is zero, meaning no probability density. On the other hand, the cumulative distribution function \( F(y) = \frac{1}{4}y^2 \) forms a parabolic curve from the origin to the point (2,1), showing how probabilities accumulate to reach 1 as \( y \) increases to 2. Graphing these functions helps in identifying key characteristics like increasing trends and can highlight visual differences, like linear versus quadratic growth. Remember, for the PDF, the area under the curve signifies probability, while for the CDF, the curve itself directly indicates total probability at each \( y \).

Calculating probabilities using both the PDF and CDF provides flexibility and deeper understanding. For example, to find \( P(1 \leq Y \leq 2) \), we can use the CDF: \( P(1 \leq Y \leq 2) = F(2) - F(1) \). Substituting the values from the CDF, \( F(2) = 1 \) and \( F(1) = \frac{1}{4} \), yields \( P(1 \leq Y \leq 2) = \frac{3}{4} \). This calculation shows the proportion of the distribution lying between these bounds. Alternatively, using integration and geometry, this probability can also be obtained by calculating the area under \( f(y) \) from 1 to 2: \( \int_{1}^{2} \frac{1}{2}y \, dy \). This approach effectively measures the weighted area between these points, reinforcing the concept of probability in the context of area under a curve. Both methods should give the same result if calculated correctly, demonstrating consistency and choice in problem-solving.