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While reading about random vectors, I learned that...

$$ E\left[\vec{X}\right] = \left[\begin{array}{cccc} E\left[\vec{X}_1\right] & E\left[\vec{X}_2\right] & \cdots & E\left[\vec{X}_m\right]\end{array}\right]^T\in\mathbb{R}^m $$

But what about for higher moment? For example purposes, suppose that...

  • $\vec{X} = \left[\begin{array}{cc}U & V\end{array}\right]^T$.
  • $P\left[U = u, V = v\right] = \cases{\frac{3}{2}x^2 + y & $0 < x,y < 1$\\ 0 & otherwise}$.

How would I find $E\left[\vec{X}^n\right]$ & what would the answer be for a small example like $E\left[\vec{X}^3\right]$?

P.S. A similar q has already been asked but I seek a complete example.

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