(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Hi all, I have some difficulty understanding the following problem, help is greatly appreciated!

Let ##U_1, U_2, U_3## be independent random variables uniform on ##[0,1]##. Find the probability that the roots of the quadratic ##U_1 x^2 + U_2 x + U3## are real.

2. Relevant equations

3. The attempt at a solution

From the determinant, the following must hold for real solutions

$$U_2 ^2 \geq 4U_1 U_3$$

And the corresponding probability to compute is

$$P(U_1 \leq \frac{U_2 ^2}{4U_3} )$$

I fixed ##u_2## which gives me the following function,

$$P(U_1 \leq \frac{u_2 ^2}{4U_3} | U_2 = u_2)$$

which I solved for by integrating over the domain specified by inequality, the result was

$$u_2 ^2 /4 - \frac{u_2 ^2}{2} \ln (u_2 / 2)$$

Now I want the 2nd equation from the top, which gives the total probability of having ##U_1 \leq \frac{U_2 ^2}{4U_3} ##. My intuition was to integrate the result from ##u_2 \in [0,1]##, which according to my solutions manual turns out to be true. But doesn't this imply that

$$P(U_1 \leq \frac{U_2 ^2}{4U_3} ) = \int P(U_1 \leq \frac{u_2 ^2}{4U_3} | U_2 = u_2) \ du_2$$

which is something I've not really seen before. But it leads to the answer, so my question is, why would this step be correct?

I have tried looking around the net for something relating a conditional CDF to a joint CDF, and I found this,

$$F(x|y) f_y (y) = \frac{dF(x,y)}{dy}$$

could it be that

$$F(x,y) = \int_{-\infty}^{y} F(x|u) f_y (u) \ du $$

$$P(X\leq x) = \lim_{y \to \infty} F(x,y) = \int_{-\infty}^{\infty} F(x|u) f_y (u) \ du $$

in my case ##f_y## would simply be 1, and so would the upper limit of my integral. Is this sound?

Many thanks in advance for any assistance!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Probability theory: Understanding some steps

Tags:

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**