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Joint density functions are crucial in probability and statistics, especially when dealing with multiple random variables simultaneously. These functions help us understand how two (or more) related random variables behave together. In this exercise, we have the random variables \(Y_1\) and \(Y_2\), which represent proportions within a chemical mixture. The given joint density function is \[ f\left(y_{1}, y_{2}\right)=\begin{cases} 2, & 0 \leq y_{1} \leq 1, 0 \leq y_{2} \leq 1, 0 \leq y_{1}+y_{2} \leq 1 \ 0, & \text{elsewhere} \end{cases}\]This function is defined over a triangular region on a coordinate plane, meaning that the values of both \(Y_1\) and \(Y_2\) lie between 0 and 1, and their sum does not exceed 1. This structure ensures that both variables are considered as proportions (parts of a whole). It is essential to accurately project this triangular region when solving problems related to joint probabilities.

Integral calculus plays a significant role when dealing with joint density functions because it allows us to calculate probabilities over specific regions. For example, probabilities of events where joint variables have certain constraints can be determined by integrating the joint density over the desired region. Here, the regions of integration are determined by inequalities like \(Y_1 \leq 3/4\) and \(Y_2 \leq 3/4\). The integration process involves setting up double integrals, which look like this for Part (a):\[\int_{0}^{3/4} \int_{0}^{1-y_{1}} 2 \, dy_{2} \, dy_{1}\]First, we integrate with respect to \(y_2\) while keeping \(y_1\) constant. Then, for each slice defined by a particular \(y_1\), we integrate across the entire valid range of \(y_2\). This method is applied iteratively across the \(y_1\) limits. Calculating this integral accurately gives the probability for the specified conditions of \(Y_1\) and \(Y_2\).

The concept of probability borders refers to the boundaries that define the region over which we're calculating probability. In joint probability distributions, these borders need considerable attention to ensure they're correctly determined. The inequalities that define these borders hold much importance, such as \(Y_1 \leq 1/2\) and \(Y_2 \leq 1/2\) for Part (b). The boundary condition, \(Y_1 + Y_2 \leq 1\), emphasizes the dependence between \(Y_1\) and \(Y_2\) as it clarifies the amount both can sum up to. Understanding and identifying the probability borders help delineate the precise area within which the integration occurs. Graphing these borders makes it visually apparent, aiding in setting up the integrals accurately and ensuring that we only calculate probabilities over the correctly defined region.