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In order statistics, the joint probability density function (joint PDF) provides a way to describe the likelihood of multiple random variables occurring together. For the given problem, we are tasked with determining the joint PDF of transformed variables \(Z_1, Z_2,\) and \(Z_3\) derived from their original versions \(Y_1, Y_2,\) and \(Y_3\). The transformation involves mapping to a new space under specified conditions.
The general process involves a transformation using the Jacobian of the function that converts the random sample \(Y\) into the sample \(Z\). You begin by defining the transformation functions: \(Z_1 = Y_1, Z_2 = Y_2,\) and \(Z_3 = Y_1 + Y_2 + Y_3\). The Jacobian—a determinant of the transformation matrix—helps to adjust the scale of the distribution when switching variables.
The next step involves substituting these transformation equations into the original joint PDF expression. Solving this, given the constraints of the transformation, delivers the joint PDF for \(Z\), providing insight on the dependencies and likelihood among \(Z_1, Z_2,\) and \(Z_3\).

Sufficient statistics are pivotal in statistical inference because they summarize the sample's information about unknown parameters without needing the entire data set. In this exercise, \(Z_1\) and \(Z_3\) are claimed to be sufficient statistics for the parameters \(\theta_1\) and \(\theta_2\).
The factorization theorem can be employed to confirm that \(Z_1\) and \(Z_3\) sufficiently summarize information about these parameters. According to the theorem, a statistic is sufficient if the joint PDF can be decomposed into a product of two functions:

• One function depending on the statistic and the parameter
• Another depending only on the sample data

For our example, once the joint PDF of \(Z_1, Z_2,\) and \(Z_3\) is obtained, we check if it can be split according to these rules. If feasible, this proves that knowing \(Z_1\) and \(Z_3\) is as informative about \(\theta_1\) and \(\theta_2\) as having the whole data set.

Mathematical transformations are utilized to convert data from one coordinate system to another. This is particularly useful in statistics where you might need to switch between variables that better reveal properties of interest. For example, in this problem, the transformation from \(Y_{coordinates}\) to \(Z_{coordinates}\) is executed through the equations \(Z_1 = Y_1\), \(Z_2 = Y_2\), and \(Z_3 = Y_1 + Y_2 + Y_3\).
Such transformations regularly involve computing the Jacobian matrix, which quantifies how much the transformation distorts areas or volumes in the space. The determinant of this matrix, known as the Jacobian determinant, modifies the probability density accordingly.
This process involves:

• Identifying the transformation functions for each variable.
• Evaluating the Jacobian matrix based on these functions.
• Using the Jacobian determinant to adjust the original PDF, ensuring that the total probability remains consistent.

By thoroughly understanding these transformations, you gain the ability to analyze the dependency and correlation characteristics between transformed variables and simplify the derivation of statistical properties like joint PDFs.