Probability that roots of quadratic are real

[iomtt6076]

Homework Statement

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.

Homework Equations

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.

 

[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?

 

[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]?

 

[vela]

Yes.

 

[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.

 

[vela]

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

 

[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.

 

[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.

 

[ysm1nz]

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.

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.