Question

Let \(X\) follow \(N(3,1)\). Then the value of \(E\big(X^4(X-3)\big)\) equals ______ (answer in integer).

Hint:

Write \(Z=X-3\sim N(0,1)\) and expand \((Z+3)^4Z\); only the even-power terms in \(Z\) survive after taking expectation.

Answer 1

Correct Answer: 144

Step 1: Standardize.
$Z=X-3\sim N(0,1)$, target $E((Z+3)^4 Z)$.

Step 2: Stein's identity.
For $Z\sim N(0,1)$: $E[Zh(Z)]=E[h'(Z)]$.

Step 3: Apply with h(Z)=(Z+3)^4.
$h'(Z)=4(Z+3)^3$, so $E[(Z+3)^4Z]=4E(X^3)$.

Step 4: Find E(X^3).
$X^3=(3+Z)^3=27+27Z+9Z^2+Z^3$; odd terms vanish, $E(Z^2)=1$: $E(X^3)=27+9=36$.

Step 5: Combine.
\[ 4\times36=144 \]

Step 6: Cross check.
Direct expansion gives same $144$.
\[ \boxed{144} \]