I am working on a problem and am a bit stuck. It is:
For X ~ $\mathcal N $(-2,2) find E(X$^3$)
What I know so far is that:
For X ~ N(0,1) EX = 0 and VarX = 1
I am confused though how we find expected values from that normal distribution. Should I use integrals to solve for this? If so, how do you set up the proper integral (do I use the gaussian equation for this?) I am guessing the integral would go from negative infinity to infinity.
Edit: this is the work I have done so far:
For X ~ $\mathcal N $(-2,2) Find E($\ X^3 $)
Z ~ $ \mathcal N $(0, 1)
X = $ \mu $ + $\sigma $X ~ $ \mathcal N $($\mu$,$\sigma^2$)
$\mu$ = -2, $\sigma^2$ = 2, $\sigma$ = $\sqrt 2$
X = -2 + $\sqrt 2$Z w/ Z ~ (0,1)
(X)$^3$ = ($\sqrt 2$Z - 2)$^3$
-12Z$^2$ + 8 (because of symmetry)
Z$^2$ = 1 so therefore E(X$^3$) = -12 + 8 = -4
Does this look correct? How would you set this up simply by definition?
Following your idea to reduce it to calculating moments of $Z\sim \mathcal{N}(0,1)$ rather than using MGFs or the definition of expectation:
If $X\sim \mathcal{N}(\mu, \sigma^2)$ then $X=\mu+\sigma Z$ where $Z\sim \mathcal{N}(0,1)$, as you know, so that $$X^3=\mu^3+3\mu^2 \sigma Z +3 \mu (\sigma Z)^2 +3(\sigma Z)^3,$$ so that upon taking the expectation, we get $$\mathbb{E}(X^3)=\mu^3+3\mu^2 \sigma \mathbb{E}(Z)+3\mu\sigma^2 \mathbb{E}(Z^2)+3 \sigma^3 \mathbb{E}(Z^3).$$ Now after simplifying with the facts that $\mathbb{E}(Z)=0$, $\mathbb{E}(Z^2)=1$ and $\mathbb{E}(Z^3)=\int_{-\infty}^\infty z^3 \phi(z)\mathrm{dz}=0$ since $\phi(z)=\frac{1}{\sqrt{2\pi}}e^{-z^2/2}$ is an even function and $z^3$ is an odd function so $z^3\phi$ is an odd function and integrating an odd function over a symmetric interval is zero, we finally obtain $E(X^3)=\mu^3+3\mu\sigma^2.$
You can check this is correct by one of the other methods, e.g. if you know $M_X(t):=\mathbb{E}(e^{tX})=e^{\mu t+0.5 \sigma^2 t^2}$ then you can compute three derivatives and verify that $M_X^{'''}(0)=\mathbb{E}(X^3)$.
Note: all odd moments of a standard normal RV are zero by the same argument involving odd/even functions and symmetry.
Hope this helps, comment for clarification.