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A joint probability distribution describes the probability of two or more random variables occurring at the same time. In the context of continuous random variables, it specifies a function called a joint probability density function (pdf), denoted as \(f(x_1, x_2)\). This function gives the likelihood of \(X_1\) and \(X_2\) taking on particular values within a specified range.
In the given problem, \(X_1\) and \(X_2\) represent the times taken to complete two tasks. The joint pdf is defined as:

• \(f(x_1, x_2) = 2(x_1 + x_2)\) for \(0 \leq x_1 \leq x_2 \leq 1\)
• And \(f(x_1, x_2) = 0\) otherwise.

This means that within the constraint \(0 \leq x_1 \leq x_2 \leq 1\), the probability density is determined by \(2(x_1 + x_2)\). It captures both the individual behaviors of \(X_1\) and \(X_2\), as well as their dependence on each other, since the time for the second task (\(X_2\)) is always greater than or equal to the time for the first task (\(X_1\)). Understanding this relationship is crucial for solving problems that involve the transformation of variables.

Transformation of Variables is a powerful technique in probability theory. It involves changing variables to simplify the analysis of a problem. In our case, transformations help us find the pdf of new variables like the total completion time and the time difference.
For the total completion time of the two tasks, we define a new variable \(Y = X_1 + X_2\). This transformation allows us to describe the combined effect of \(X_1\) and \(X_2\) as a single random variable. Similarly, we define \(Z = X_2 - X_1\), representing the difference in times, which helps to understand the variability between the tasks.
The transformation process involves setting up equations for new variables and finding their ranges. In the exercise:

• For \(Y\), the total time ranges from 0 to 2 because \(X_1\) and \(X_2\) can independently vary from 0 to 1.
• For \(Z\), its range is from 0 to 1 given that the second task cannot be completed in less time than the first.

Performing such transformations simplifies the calculations required to find new pdfs. It helps recast complex relationships into manageable forms.

The Probability Density Function (pdf) is a critical concept when dealing with continuous random variables. It represents the density of probability at any given point for a random variable. For a function \(f(x)\) to be a probability density function, it must satisfy the condition \(\int_{-\infty}^{\infty} f(x)\,dx = 1\).
Occupying center stage in our analysis is the derivation of pdfs for the transformed variables \(Y\) and \(Z\). By performing transformations:

• For \(Y = X_1 + X_2\), its pdf is derived by differentiating the cumulative distribution function (CDF) \(F_Y(y)\). The resulting pdf is: \[f_Y(y) = \begin{cases} y, & 0 \leq y \leq 1 \ 2-y, & 1 < y \leq 2 \ 0, & \text{otherwise} \end{cases}\]
• For the difference \(Z = X_2 - X_1\), the pdf is found similarly with: \[f_Z(z) = \begin{cases} 2(1-z)z, & 0 \leq z \leq 1 \ 0, & \text{otherwise} \end{cases}\]

The pdfs enable us to understand how probability is distributed over time durations, providing deeper insights into task completion and the variability of event outcomes.

The Cumulative Distribution Function (CDF) of a continuous random variable is the probability that the variable takes on a value less than or equal to \(x\). For a random variable \(Y\), the CDF \(F_Y(y)\) is the integral of its pdf from \(-\infty\) to \(y\).
In practical terms, the CDF offers a way to model cumulative probabilities and is especially useful when deriving pdfs for transformed variables. In the current exercise:

• The CDF for \(Y = X_1 + X_2\) is expressed as: \[F_Y(y) = \int_{0}^{\min(1, y/2)} \int_{x_1}^{\min(1, y-x_1)} 2(x_1+x_2)\, dx_2\, dx_1\]
• It is differentiated to obtain the pdf \(f_Y(y)\).

Similarly:

• For the difference \(Z = X_2 - X_1\), integration to find \(F_Z(z)\) and differentiating yields the pdf \(f_Z(z)\).

CDFs act as building blocks in probability theory, allowing us to transition from cumulative probabilities to specific probability densities, thus formulating the required distributions effectively.