So given a continuous random variable $X$ with density function $a + bx^2$ for $0\le x \le 1$ and $0$ otherwise.
If any values can be assigned to $a$ and $b$, can the distribution take on any expected value?
My immediate thought were that yes because there would exist a combination of $a$ and $b$ that would yield any mean. I then remembered that the PDF must always be non-negative and that the CDF must equal $1$ as $X$ approaches infinity. This caused me to doubt my initial instincts and I now believe that there are likely values that the conditions for the CDF or PDF would not be met. I would be interested to hear your reasoning on the problem.