Page 150 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 41 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 143

We assume that r 2 0 and 0 < 8 < 2x. With this assumption, the transformation

has the inverse transformation
x=rcos8 y=rsin0
Since

by Eq. (4.23) we obtain
fRe(r, 8) = rfxu(r cos 0, r sin 8) (4.90)

4.28. A voltage V is a function of time t and is given by
V(t) = X cos ot + Y sin ot
in which o is a constant angular frequency and X = Y = N(0; a2) and they are independent.
(a) Show that V(t)may be written as
V(t) = R cos (ot - 0) (4.92)

(b) Find the pdfs of r.v.'s R and O and show that R and O are independent.
(a) We have
V(t) = X cos wt + Y sin wt
)
= ( ,& cos cut + Y sin at
, X2+ Y
= ,/n(cos 0 cos at + sin 0 sin at)
= R cos(at - 0)
Y
and
where R=JW' @=tan-'-
X
which is the transformation (4.89).
(b) Since X = Y = N(0; a2) and they are independent, we have
-(xZ + yz)/(2az)
fxh Y) = 2no' e
Thus, using Eq. (4.90), we get

Now

and fa&, 8) = fR(r) fe(8); hence, R and 0 are independent.
Note that R is a Rayleigh r.v. (Prob. 2.23), and 0 is a uniform r.v. over (0, 271.).