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Given $\mathbb E(5X+2)=12$ and $\mathbb E(X|Y)=Y^3$ compute $\mathbb E(Y^3)$.

I've been trying this a million different ways and can't seem to reach a final answer, I would love any suggestions!

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2 Answers 2

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$$\mathbb E(5X+2)=5\,\mathbb E(X)+2=12 \implies \mathbb E(X)=2$$

$$\mathbb E(Y^3)=\mathbb E\bigl(\mathbb E(X|Y)\bigr)=\mathbb E(X)=2$$

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    $\begingroup$ For any readers: Reference this answer to see that $\mathbb E\bigl(\mathbb E(X|Y)\bigr)=\mathbb E(X)$. $\endgroup$ Mar 27, 2019 at 21:13
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$E[X] = 2$.

Since $E[ X|Y] = Y^3$, then $E[Y^3] = E[ E[X|Y]] = E[X] = 2$.

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