Proof of PDF Integrals Hi guys my professor gave us some sample proofs to try at home and I was having trouble with 4 of them. I figured out how to do part (a) by using polar coordinates but cannot wrap my head around the other 4. We are to try and finish these by hand so no calculators could be used.

Any help will be appreciated! Thanks!
 A: Once you have the first integral, the others are straightforward.  I will use $\mu$ instead of $\langle x \rangle$, in accordance with statistical notation.  I will also denote for convenience $$f(x;\mu,\sigma) = \frac{1}{\sqrt{2\pi}\sigma} e^{-(x-\mu)^2/(2\sigma^2)}.$$  Then consider the transformation $z = \frac{x-\mu}{\sigma}$: $$\begin{align*} \int_{x=-\infty}^\infty x f(x;\mu,\sigma) \, dx &= \sigma \int_{x=-\infty}^\infty \frac{x-\mu}{\sigma} f(x;\mu,\sigma) \, dx + \mu \int_{x=-\infty}^\infty f(x;\mu,\sigma) \, dx \\ &= \sigma \int_{z=-\infty}^\infty z f(z;0,1) \, dz + \mu \\ &= \mu,  \end{align*}$$  because $zf(z;0,1)$ is an odd function, so its integral is zero over the entire real line.
For (c), we have $$\int_{x=-\infty}^\infty (x-\mu)^2 f(x;\mu,\sigma) \, dx = \sigma^2 \int_{z=-\infty}^\infty z^2 f(z;0,1) \, dz.$$  Apply integration by parts with the choice $u = z$, $du = dz$, $dv = z f(z;0,1) \, dz$, $v = -f(z;0,1)$.
For (d), note again that the standardizing transformation results in an integrand that is an odd function, so its integral is zero.
For (e), we use integration by parts as in (c).
I should also point out that the easiest way to do these integrals is to use the moment generating function:  Calculate $$M_X(t) = {\rm E}[e^{tX}] = \int_{x=-\infty}^\infty e^{tX} f(x;\mu,\sigma) \, dx = e^{\mu t + \sigma^2 t^2/2}.$$  Then the successive raw moments are given by $${\rm E}[X^k] = \frac{d^k}{dx^k}\left[M_X(t)\right]_{t=0}.$$
