Problem 3.5 (Geometric derivation of Q function bound) Let x_(1) and x_(2)
denote independent standard Gaussian random variables.
(a) For a>0, express P[|x_(1)|>a,|x_(2)|>a] in terms of the Q function.
(b) Find P[x_(1)^(2)+x_(2)^(2)>2a^(2)].
Hint Transform to polar coordinates. Or use the results of Problem 3.4.
(c) Sketch the regions in the (x_(1),x_(2)) plane corresponding to the events con-
sidered in (a) and (b).
(d) Use (a)-(c) to obtain an alternative derivation of the bound Q(x)<=(1)/(2)e^(-(x^(2))/(2))
for x>=0 (i.e., the bound in Problem 3.3(a)).
Problem 3.5 (Geometric derivation of Q function bound) Let X, and X, denote independent standard Gaussian random variables.
(a) For a > 0, express P[|X|> a, |X,|> a] in terms of the Q function. (b) Find P[X?+X2> 2a2].
Hint Transform to polar coordinates. Or use the results of Problem 3.4.
(c) Sketch the regions in the (x,, x,) plane corresponding to the events con- sidered in (a) and (b).
for x > 0 (i.e., the bound in Problem 3.3(a)).