<P> This integral is over the half - plane which lies under the line x+y = z . </P> <P> The key observation is that the function </P> <Dl> <Dd> f (x) g (y) = (1 / 2 π) e − (x 2 + y 2) / 2 (\ displaystyle f (x) g (y) = (1 / 2 \ pi) e ^ (- (x ^ (2) + y ^ (2)) / 2) \,) </Dd> </Dl> <Dd> f (x) g (y) = (1 / 2 π) e − (x 2 + y 2) / 2 (\ displaystyle f (x) g (y) = (1 / 2 \ pi) e ^ (- (x ^ (2) + y ^ (2)) / 2) \,) </Dd>

Is the sum of two independent random variables independent