Probability that roots of quadratic are real

1. Jan 17, 2010

iomtt6076

1. The problem statement, all variables and given/known data
Let U1, U2, and U3 be independent random variables uniform on [0,1]. Find the probability that the roots of the quadratic U1x2+U2x+U3 are real.

2. Relevant equations

3. The attempt at a solution
So we need to find P(U22>4U1U3), which involves evaluating some integral. The think the integrand would be 1 since we are dealing with uniform random variables. But beyond that, I need assistance in figuring out whether it should be a double integral or a triple integral and the limits of integration.

2. Jan 17, 2010

Dick

It's a triple integral, sure. But it's pretty straightforward. Integrate U1 and U3 from 0 to 1. That makes your only integral with nontrivial limits the U2 integral. What are the limits there?

3. Jan 17, 2010

iomtt6076

Are we trying to find the volume above the surface defined by $$U_2=2\sqrt{U_1U_3}$$ and inside [0,1]x[0,1]x[0,1]?

4. Jan 17, 2010

vela

Staff Emeritus
Yes.

5. Jan 17, 2010

iomtt6076

Why isn't it

$$\int_0^1\int_0^1\int_{2\sqrt{U_1U_3}}^1 dU_2\,dU_1\,dU_3$$

I can't figure out why my limits of integration for U2 are wrong.

6. Jan 17, 2010

vela

Staff Emeritus
Here's a hint: What's the lower limit equal to when U1=U3=1?

7. Jan 17, 2010

Dick

Oh, heck. Thanks, vela. Requiring U2<=1 does create restrictions on the range of both U1 and U3. Backtrack and fix my stupid suggestion of integrating both U1 and U3 from 0 to 1 by requiring each variable in turn be less than or equal to 1. Sorry.

8. Jan 17, 2010

iomtt6076

Thanks to both of you for your help; I finally got it. I found that drawing a picture by taking slices through the U2-axis was also essential.

9. Mar 2, 2011

ysm1nz

Would it have killed you to post the solution? I can't figure out what the limits should be. I figured U1<1/(4U3) and U3<1/(4U1) but using those as the upper limits I get nowhere.