To calculate the probability density function (PDF) of a well-defined joint function, you need to consider the joint distribution of the variables involved. The PDF represents the likelihood of obtaining a specific combination of values for the variables.

Here's a step-by-step guide to calculating the PDF of a well-defined joint function:

1. Identify the variables: Determine the variables involved in the joint function. For example, let's consider a joint function of two variables, X and Y.

2. Determine the joint distribution: Find the joint distribution function, which describes the probability of obtaining specific combinations of values for X and Y. This distribution is often denoted as f(x, y).

3. Differentiate the joint distribution: Calculate the partial derivatives of the joint distribution function with respect to each variable. For our example, this would involve finding ∂²f(x, y)/∂x∂y.

4. Obtain the PDF: The PDF is obtained by taking the absolute value of the partial derivatives and normalizing them. Normalize the derivatives by dividing them by the appropriate scaling factor. This ensures that the PDF integrates to 1 over the entire range of the variables.

5. Express the PDF: Write the PDF as a function of the variables. For our example, the PDF would be denoted as f(x, y).

6. Calculate probabilities: Once you have the PDF, you can use it to calculate probabilities associated with specific events or ranges of values. This involves integrating the PDF over the desired range. For example, to find the probability that X and Y fall within certain intervals, you would integrate the PDF over those intervals.

It's worth noting that the specific method for calculating the PDF of a joint function may vary depending on the distribution and the variables involved. For commonly encountered distributions, such as the multivariate normal distribution, there are established formulas and techniques available.

If you want to dive deeper into the topic, I recommend referring to textbooks or online resources on probability theory and mathematical statistics. Some references that may be helpful include:

• "Probability and Statistical Inference" by Robert Hogg and Elliot Tanis
• "Mathematical Statistics with Applications" by Dennis Wackerly, William Mendenhall, and Richard L. Scheaffer
• "Introduction to Probability Models" by Sheldon M. Ross

Remember to consult these references for more detailed examples and explanations tailored to your specific needs.