Expected Value – How to Calculate Expected Value of (3X – 5X^2 + 1)

Question

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.

Answer 1

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

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

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