What is the probability that the polynomial has real roots? asked for the probability that the quadratic polynomial $ax^2+bx+c$ has real roots if the parameters $a,b,c$ are exponentially distributed with common parameter. (The parameter was specified as $1$ but doesn't matter.)

This didn't seem to me like a natural question to ask, since the exponential distribution has nice properties with respect to addition and not so much with respect to multiplication, and the discriminant condition $b^2\ge4ac$ is multiplicative; so I was a bit surprised by the simple answer $\frac13$. This seems to cry out for a symmetric proof, e. g. using a transformation to three equivalent variables one of which must equiprobably be the greatest to fulfill the condition. However, I couldn't come up with such a transformation (I tried $\sqrt{ab}$, $\sqrt{bc}$, $\sqrt{ca}$ and $a-b$, $b-c$, $c-a$) and could only obtain the probability by performing an asymmetric and somewhat involved triple integration.

Can you provide a more elegant proof of this result?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.