1.7 (

) www In this exercise, we prove the normalization condition (1.48) for the univariate Gaussian. To do this consider, the integral I =

 ∞

−∞

exp



− 1 2σ2 x2



dx (1.124)

which we can evaluate by first writing its square in the form I2 =

 ∞

−∞

 ∞

−∞

exp



− 1 2σ2 x2 − 1 2σ2 y2



dx dy. (1.125)

Now make the transformation from Cartesian coordinates (x, y) to polar coordinates

(r, θ) and then substitute u = r2. Show that, by performing the integrals over θ and u, and then taking the square root of both sides, we obtain I =

2πσ21/2

. (1.126)

Finally, use this result to show that the Gaussian distribution N(x|μ, σ2) is normalized.