I have a question regarding linear combinations/transformations in statistics. I'm quite sure the answer is relatively easy, but I can't seem to find a solution that corresponds to my solution manual.
The question is as follows: Say X is a uniform random variable on the interval $]0,2[$. Assume the following random variable $Q=6X^2+3$. What is the expected value of $Q$?
I started out by transforming the equation for $Q$. $$E[Q] = E[6X^2+3]$$ $$= 6E[X^2] +3$$ $$ = 6(E[X])^2 + 3 + \frac{1}{2}Var(X)*2$$ Using the following formula from my 'cheat sheet': $E[g(X)] \approx g(E[X]) + \frac{1}{2}Var(X)*g''(E[X])$
Now I'm kind of stuck because I'm not sure if this approach is correct.
Any tip would be very helpful! This was translated from Dutch to English, so keep in mind that there might be some errors regarding statistic jargon.