If $E[X]=2, E[X^2]=5, E[X^3]=0$, and $E[X^4]=30$, find $E[(X-\pi)^3]$.
I keep getting a negative number when I do this out! Please help! Here's my work:
$E[(X-\pi)^3]=E[X^3]-3\pi E[X^2]+3\pi^2 E[X] - E[\pi^3]$
$E[(X-\pi^3)]=0-15\pi+6\pi^2-\pi^3 \approx -19$
The expression $(X-\pi)^3$ can be negative, so its expectation can be negative. As a simpler example, $E[X-\pi] = 2-\pi$ is also negative.