# Expectation of roots of a quadratic equation

The quadratic equation $x^2 -ax+ b = 0$ is known to have two real roots, $X_1$ and $X_2$ $(X_1 > X_2)$ but the coefficient $b$ is a positive unknown and can be assumed to have a uniform distribution in the permissible range of variation. Then what is the expected value of $X_1$.

I found that the roots have terms $a$ and $b$ in it. I took its expectation. But nothing is known about $a$ and it is just mentioned that $b$ follows uniform distribution but range is not specified. So how do l find the expectation.

• Is this a self-study question? If so, please read the tag wiki and update the question to say what you have tried and where you are stuck. – GeoMatt22 Apr 4 '17 at 13:27
• GeoMatt2 simply means that you add the self-study tag to the tag list. – Michael Chernick Apr 4 '17 at 13:37
• @MichaelChernick yes, forgot that part! Also, instead of saying "the roots have terms ...", write out the equations (and all your work so far). As for the question, my suggestion would be to take $b\in[b_1,b_2]$, and consider first the $a=0$ case. – GeoMatt22 Apr 4 '17 at 13:40
• Use the discriminant of the quadratic to find the range of variation for $b$. Now, use the fact that $X_1$ satisfies the quadratic equation and take expectation. Now, you can solve for expected value as a quadratic equation. – TenaliRaman Apr 4 '17 at 13:44