What is the probability that the polynomial has real roots? The variables $a$, $b$ and $c$ are iid random variables with exponential distribution with common parameter $1$. What is the probability that the polynomial $ax^2+bx+c$ has real roots?
My attempt:
$\Pr[ax^2+bx+c\;\text{has real roots}]=\Pr[b^2-4ac\geqslant0]=\Pr[b\geqslant2\sqrt{ac}\;\text{or}\;b\leqslant-2\sqrt{ac}]=\Pr[b\geqslant2\sqrt{ac}]=\int_0^\infty\int_0^\infty\int_{2\sqrt{ac}}^\infty f_{a,b,c}(a,b,c)dadbdc$
Am I right? How can I continue?
 A: \begin{align}
\mathsf P\left(a\gt \frac{b^2}{4c}\right)
&=\int_0^\infty\int_0^\infty\int_{\frac{b^2}{4c}}^\infty\exp(-a-b-c)\,\mathrm da\,\mathrm db\,\mathrm dc\\
&=\int_0^\infty\int_0^\infty\exp\left(-\frac{b^2}{4c}-b-c\right)\mathrm db\,\mathrm dc\\
&=\int_0^\infty\int_0^\infty c\exp\left(-\frac{s^2}4c-sc-c\right)\mathrm ds\,\mathrm dc\\
&=\int_0^\infty\int_0^\infty c\exp\left(-\left(\frac s2+1\right)^2c\right)\mathrm dc\,\mathrm ds\\
&=\int_0^\infty\left(\frac s2+1\right)^{-4}\,\mathrm ds\\
&=16\int_2^\infty s^{-4}\,\mathrm ds\\
&=\frac23
\end{align}
(where I used the substitution $b=sc$), so the desired probability is $\frac13$. As I wrote in a comment, I'm surprised that this has such a nice closed form, even more so that the probability is $\frac13$, which cries out for a solution with a transformation to three symmetric quantities of which one must be the greatest; I've posted a new question about this: Symmetric proof for the probability of real roots of a quadratic with exponentially distributed parameters.
