Statistics: Odd Moments

Question

Need help with this stat question. I know you start by integrating $z^k f(z)$ from $-\infty$ to $0 +$ integral of $z^k f(z)$ from $0$ to $\infty$. After that I'm stuck.

Answer 1

Hint 1: Show that if $x\mapsto f(x)$ is even, then $x\mapsto x^kf(x)$ is odd if $k$ is an odd positive integer.

Hint 2: What is the net signed area under the graph of an odd function over an interval centered at $0$?

Answer 2

The $k$th moment is $$ \int_{-\infty}^\infty z^k \varphi(z)\,dz $$ where $$ \varphi(z) = \frac 1 {\sqrt{2\pi}} e^{-z^2/2}. $$

If $k$ is odd then we have $(-z)^k = -(z^k)$. We also have $(-z)^2 = z^2$, so that $\varphi(-z)=\varphi(z)$. Then let $u=-z$ so that $-du=dz$, and we have \begin{align} \int_{-\infty}^\infty z^k \varphi(z)\,dz & = \int_{-\infty}^0 z^k \varphi(z)\,dz + \int_0^\infty z^k \varphi(z)\,dz \\[10pt] & = \int_\infty^0 (-u)^k \varphi(-u)\,(-du) + \int_0^\infty z^k \varphi(z)\,dz \\[10pt] & = \int_\infty^0 u^k \varphi(u)\,du + \int_0^\infty z^k \varphi(z)\,dz \\[10pt] & = \int_\infty^0 z^k \varphi(z)\,dz + \int_0^\infty z^k \varphi(z)\,dz \\[10pt] & = 0. \end{align}