L10.4 Total Probability & Total Expectation Theorems  Summary and Q&A
TL;DR
Conditional PDFs and expected value in continuous random variables have analogous properties to their discrete counterparts, and can be derived using integrals and densities.
Questions & Answers
Q: What are the properties of conditional PDFs in the continuous case?
Conditional PDFs in the continuous case have analogous properties to conditional PMFs in the discrete case. They are derived using integrals and densities, with sums replaced by integrals and PMFs replaced by PDFs.
Q: How is conditional expectation defined for continuous random variables?
Conditional expectation for continuous random variables is defined similar to the discrete case, but using the corresponding conditional PDF. It involves taking the integral of the random variable multiplied by the conditional PDF.
Q: What is the total expectation theorem in the continuous case?
The total expectation theorem in the continuous case involves taking a weighted average of conditional expectations based on the values of the density. Under each possibility of the conditioning variable, the expected value of the other random variable is calculated and multiplied by the corresponding density value.
Q: How are expected value rules calculated in the continuous case?
In the continuous case, expected value rules are calculated using integrals and densities. For example, to calculate the expected value of a function of a random variable conditioned on another random variable, the function is multiplied by the conditional PDF and integrated.
Summary & Key Takeaways

Conditional PDFs in the continuous case have similar properties to conditional PMFs in the discrete case, with the sum replaced by an integral and PMFs replaced by PDFs.

The conditional expectation for continuous random variables can be defined in a similar manner to discrete random variables, using the corresponding conditional PDF.

The total expectation theorem in the continuous case involves taking a weighted average of conditional expectations based on the values of the density.