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

4.10 TRANSFORMATIONS OF RANDOM VARIABLE FAMILIES

In this section, we summarize transformation results for some of the parametric distributions discussed in Chapter 3. Derivations are provided only for a few of the transformations using the previous techniques; others are considered in the problems at the end of this chapter.

4.10.1 Gaussian Transformations

Let be iid Gaussian random variables with parameters . The results for several transformations are provided in Table 4.3.

TABLE 4.3 Transformations of Gaussian Random Variables

Transformation	Conditions on iid {xn}	Random Variable Y {parameters}
Y = exp(X)	None	Log-normal
Y = |X|		Half-normal
Y = X1/X2	,	Cauchy