I have an exercise where I need to find the expected value of $3X – 5X^2 + 1$, and I don't know where start. I know that $E(aX + b) = a cdot E(X) + b$, and also that $Var(X) = E(X^2) – [E(X)]^2$, but I don't know how to use these formulas. Can I have a hint on how to start? I don't need to know the PDF, but only try to expand the problem using the formulas I know.

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#### Best Answer

Given a unknown disttibution with $E(X)=mu$ and $V(x)=sigma^2$:

$$E(3X-5X^2+1)=3E(X)-5E(X^2)+E(1)$$ $$E(3X-5X^2+1)=3E(X)-5[V(x)+E(X)^2]+E(1)$$ $$E(3X-5X^2+1)=3mu-5[sigma^2+mu^2]+1$$ $$E(3X-5X^2+1)=3mu-5sigma^2-5mu^2+1$$

I consider the full expression of **@corey979** is the answer but it needs $mu$ and $sigma^2$.

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