1. Problem 10.24 In order to prove that the normal probability density func tion integrates to 1 over the interval (−∞,∞), evaluate the integral I =

∞

−∞e−1 2 x2 dxforthestandardnormaldensity.Bychangingtopolarcoordinates inthedoubleintegral I2 = ∞

∞

−∞

2

−∞e−1

(x2+y2) dxdy,verifythat I = √

2π (the polar coordinates r and θ satisfy x = r cos(θ) and y = r sin(θ) with dxdy =

rdrdθ). Also, verify that the change of variable t = 1 2

x2 in I = ∞

leads to (1 2

) = √

π.