91Ó°ÊÓ

The concept of a joint probability density function (PDF) is essential when dealing with two or more random variables. It allows us to describe the probability of these variables occurring together within a certain range. When dealing with continuous random variables like in this exercise, the joint PDF, denoted as \(f(y_1, y_2)\), represents the likelihood of the variables \(Y_1\) and \(Y_2\) assuming particular values simultaneously. In our scenario, the joint PDF is given as: \[ f(y_1, y_2) = \begin{cases} 6y_1^2y_2, & \text{if } 0 \leq y_1 \leq y_2, y_1 + y_2 \leq 2 \ 0, & \text{otherwise.} \end{cases} \]This piecewise function shows how the probability is distributed across the range of \(y_1\) and \(y_2\). The condition \(0 \leq y_1 \leq y_2, y_1 + y_2 \leq 2\) defines the domain where the probability is non-zero. Hence, it helps to analyze the dependency between \(Y_1\) and \(Y_2\) by investigating how changes in one variable affect the joint probability behavior.

To explore the dependency between random variables \(Y_1\) and \(Y_2\), we need to derive their marginal densities. These densities are function forms achieved by integrating the joint PDF with respect to the opposite variable. This practice helps to condense the joint behavior into individual variables’ behaviors. For our case, the marginal density \(f_1(y_1)\) is found by integrating over \(y_2\):\[ f_1(y_1) = \int_{y_1}^{2-y_1} 6y_1^2y_2 \, dy_2 \]Upon solving, we get:\[ f_1(y_1) = 12y_1^2(1 - y_1), \quad 0 \leq y_1 \leq 1 \]Similarly, for \(f_2(y_2)\), we integrate over \(y_1\):\[ f_2(y_2) = \int_{0}^{y_2} 6y_1^2y_2 \, dy_1 \]Resulting in:\[ f_2(y_2) = 2y_2^4, \quad 0 \leq y_2 \leq 2 \]These expressions allow us to examine each variable's distribution independently, aiding in further analysis of their probabilistic independence.

In probability theory, two random variables are termed independent if the probability of their joint occurrence equals the product of their individual probabilities. Mathematically, independence implies:\[ f(y_1, y_2) = f_1(y_1) \cdot f_2(y_2) \]Analyzing our example, substituting the computed marginal densities, we attempt to construct:\[ f_1(y_1) \cdot f_2(y_2) = 12y_1^2(1-y_1) \cdot 2y_2^4 = 24y_1^2(1-y_1)y_2^4 \]This expression does not equal the original joint density function \(6y_1^2y_2\). Therefore, as they can’t be expressed directly as a simple product of two separate functions, \(Y_1\) and \(Y_2\) are indeed dependent. One variable's value affects the probability distribution of the other, demonstrating lack of statistical independence.

Integration is a powerful tool in probability, particularly when dealing with continuous random variables. It allows us to find probabilities over certain ranges and marginal densities, which describe the probability behavior of each variable individually. In our context, integration is used to derive the marginal densities from the joint probability density function, effectively collapsing it over one variable. For example, to obtain \(f_1(y_1)\), we integrate the joint density \(f(y_1, y_2)\) over \(y_2\):\[ f_1(y_1) = \int_{y_1}^{2-y_1} 6y_1^2y_2 \, dy_2 \]This process simplifies down to a simple polynomial in \(y_1\), enabling us to understand \(Y_1\)'s behavior in isolation. Similarly for \(f_2(y_2)\), by integrating over \(y_1\), we can describe \(Y_2\)’s independent distribution. Thus, integration serves as a critical operation in probability calculations, transforming joint behaviors into individual characterizations.