Page 65 - Schaum's Outlines - Probability, Random Variables And Random Processes

5 8 RANDOM VARIABLES [CHAP 2

Fig. 2-18

2.21. Show that the pdf of a normal r.v. X given by Eq. (2.52) satisfies Eq. (2.22).
From Eq. (2.52),

Let

Then

Letting x = r cos 9 and y = r sin 9 (that is, using polar coordinates), we have

Thus,

and

2.22. Consider a function

Find the value of a such that f (x) is a pdf of a continuous r.v. X.