- #1
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Homework Statement
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.
Homework Equations
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!