Modify the program of Example 3.9 to approximate the volume beneath the bivariate standard normal density surface and above two additional
regions of integration as specified below. Use both the Riemann and Monte
Carlo methods in parts (a) and (b), with m = 10 000.
a) Evaluate P{0 <><>
0.3413452 = 0.116516. For each method, say whether it would have been
better to use m = 10 000 points to find P{0 <>
the answer.
b) Evaluate P{Z2
1 + Z2
2 <>
area 1, so remember to multiply by an appropriate constant. Because
Z2
1 + Z2
2 ∼ CHISQ(2), the exact answer can be found with pchisq(1, 2).
c) The joint density function of (Z1, Z2) has circular contour lines centered
at the origin, so that probabilities of regions do not change if they are
rotated about the origin. Use this fact to argue that the exact value of
P{Z1 + Z2 <>
with (pnorm(1/sqrt(2)) - 0.5)^2.