Page 163 - Probability, Random Variables and Random Processes

CHAP. 4) FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 155

Thus the characteristic function of N(p; a2) is obtained by setting t = jo in Mx(t); that is,

4.61. Let Y = ax + b. Show that if Yx(w) is the characteristic function of X, then the characteristic
function of Y is given by

By definition (4.50),

4.62. Using the characteristic equation technique, redo part (b) of Prob. 4.16.
Let Z = X + Y, where X and Y are independent. Then

Applying the convolution theorem of the Fourier transform (Appendix B), we obtain

THE LAWS OF LARGE NUMBERS AND THE CENTRAL LIMIT THEOREM
4.63. Verify the weak law of large numbers (4.58); that is,
lirnP(Ix,-pl>~)=O forany~
n+ do
1
where X,, = - (XI + . . + Xn) and E(XJ = p, Var(Xi) = 02.
n
Using Eqs. (4.1 08) and (4.1 1 Z), we have
c2
E(X,) = p and Var(,Yn) = -
Then it follows from Chebyshev's inequality [Eq. (2.97)] (Prob. 2.36) that

Since limn,, 02/(ne2) = 0, we get
lim P((R,-pi>&) =O
n+ w

4.64. Let X be a r.v. with pdff,(x) and let X,, ..., X, be a set of independent r.v.'s each withpdff,(x).
Then the set of r.v.'s XI, . . . , X, is called a random sample of size n of X. The sample mean is
defined by