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The concept of a joint density function is central when dealing with two or more random variables. In our scenario with \( Y_1 \) and \( Y_2 \), which represent proportions of time spent by two employees on tasks, the joint density function \( f(y_1, y_2) \) essentially gives us a way to find probabilities for these random variables occurring together.
For \( 0 \leq y_1 \leq 1 \) and \( 0 \leq y_2 \leq 1 \), the function is defined as \( f(y_1, y_2) = y_1 + y_2 \). Outside this range, it becomes 0. This function allows us to understand how the two variables interact and exist together within the given range.
Knowing the joint density helps in determining the likelihood of different scenarios involving both random variables simultaneously. For example, you can calculate the probability that both employees will simultaneously spend certain proportions of time on their tasks by integrating over the specified range.

To gain insights into each individual variable, we derive the marginal density functions from the joint density. This involves integrating the joint density with respect to one of the variables, effectively 'ignoring' the other.
For \( Y_1 \), the marginal density \( f_{Y_1}(y_1) \) is found by integrating \( f(y_1, y_2) \) over all values of \( y_2 \):

• Integrate: \( f_{Y_1}(y_1) = \int_0^1 (y_1 + y_2) \, dy_2 \)
• Result: \( y_1 + \frac{1}{2} \)

This marginal density tells us the probability distribution of \( Y_1 \) on its own.
Similarly, for \( Y_2 \), we follow the same procedure by integrating the joint density over \( y_1 \):

• Integrate: \( f_{Y_2}(y_2) = \int_0^1 (y_1 + y_2) \, dy_1 \)
• Result: \( \frac{1}{2} + y_2 \)

Marginal densities are crucial to understanding the individual behavior of random variables when they form part of a larger system.

Determining the independence of random variables \( Y_1 \) and \( Y_2 \) is essential when modeling related data points. Two random variables are independent if the joint density function can be decomposed into the product of their marginal densities; mathematically, \( f(y_1, y_2) = f_{Y_1}(y_1) \cdot f_{Y_2}(y_2) \).
The derived marginals were \( f_{Y_1}(y_1) = y_1 + \frac{1}{2} \) and \( f_{Y_2}(y_2) = \frac{1}{2} + y_2 \). Let’s compute \( (y_1 + \frac{1}{2})(\frac{1}{2} + y_2) \):

• Result: \( \frac{y_1}{2} + y_1 y_2 + \frac{y_2}{2} + \frac{1}{4} \)

This expression does not match with the given \( y_1 + y_2 \), indicating the variables are not independent.
Understanding independence helps in predicting whether altering one variable would affect the other, which is critical in statistical analysis and decision-making. Thus, knowing \( Y_1 \) and \( Y_2 \) are dependent suggests that changes in one variable may influence the other.