Page 162 - Probability, Random Variables and Random Processes

154 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS [CHAP. 4

CHARACTERISTIC FUNCTIONS
4.57. The r.v. X can take on the values x, = - 1 and x2 = + 1 with pmfs p,(x,) = p,(x2) = 0.5.
Determine the characteristic function of X.

By definition (4.50), the characteristic function of X is
Yx(o) = 0.5e-jw + 0.5d0 = i(dw + e-jw) = cos o

4.58. Find the characteristic function of a Cauchy r.v. X with parameter a and pdf given by

By direct integration (or from the Table of Fourier transforms in Appendix B), we have the following
Fourier transform pair :

Now, by the duality property of the Fourier transform, we have the following Fourier transform pair:

or (by the linearity property of the Fourier transform)
a
++ e-alwl
n(x2 + a2)
Thus, the characteristic function of X is
Yx(o) = e-alwl (4.1 24)
Note that the moment generating function of the Cauchy r.v. X does not exist, since E(Xn) + co for n 2 2.

4.59. The characteristic function of a r.v. X is given by

Find the pdf of X.
From formula (4.51), we obtain the pdf of X as

4.60. Find the characteristic function of a normal r.v. X = N(p; a2).

The moment generating function of N(p; a2) is [Eq. (4.121)]
Mx(t) = eP' + 02'2/2