B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

422 FUNDAMENTALS OF PROBABILITY THEORY
Because P(n > Ap) is the shaded area in Fig. 8.12b to the right of Ap, from Eq. (8.40c)
[ with m = O] it follows that
P(E IO) = Q (; : )
(8.42a)
Similarly,
P(E l l) = P(n < -A p
)
= Q (: : ) = P(E IO)
(8.42b)
and
P e = L P(E, m;)
= L P(m;)P(E lm;)
i
= Q (Ar) L P(mi)
O'n
(8.42c)
The error probability P e can be found from Fig. 8.12d.
Joint Distribution
For two RVs x and y, we define a CDF F x y
(x, y) as follows:
Fx y
(x, y) :@: P(x :S x and y :S y)
(8.43)
and the joint PDF P xy (x, y) as
a2
P x y (X, y) = ox oy F x y
(X, y)
(8.44)
Arguing along lines similar to those used for a single variable, we can show that as t..x ---c> 0
and t..y ---c> 0
P xy (X, y)t..x t..y = P(x < X :": X + t..x, y < y :": y + t..y) (8.45)
Hence, the probability of observing the variables x in the interval (x, x + t..x) and y in the
interval (y, y + t..y) jointly is given by the volume under the joint PDF P xy (x, y) over the
region bounded by (x, x + t..x) and (y, y + t..y), as shown in Fig. 8.13a.
From Eq. (8.45), it follows that
P(x1 < x S x2, YI < y S Y2) =
1xz 1Y2
XJ
YI
P xy
(x, y) dx dy
(8.46)
Thus, the probability of jointly observing x in the interval (x1, x2) and yin the interval (y1 , y2)
is the volume under the PDF over the region bounded by (x1 , x2) and (y1, Y2).