Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

4.8 RANDOM VARIABLE TRANSFORMATIONS

The families of random variables discussed in Chapter 3 are important for many applications. Often, their pdfs are transformed, either because of some physical mechanism (like a communication channel) or by intentional manipulation to generate variables that have more useful properties. We consider the following three basic types of transformations:

• One random variable is a function of another random variable Y = g(X), where the mapping is known and nonrandom.
• Different functions map several random variables to another set of random variables.
• A time-indexed set of random variables representing a random process are filtered by a system to generate a modified random process.

Usually, we are interested in how the pdfs of the random variables are changed by such transformations. The third case is covered in Chapter 8 when systems operating on random processes are discussed, and where we are also interested in how the correlation is affected by filtering. In this section, we focus on the first case involving a single random variable:

(4.95)

where is a memoryless mapping. Some special mappings of multiple random variables for the second case are ...