Both roots $x_1$ and $x_2$ of a square equation $x^2+ax+b=0$ can take any value from $-1$ to $1$ with equal probability . Determine the probability densities for coefficients $a$ and $b$.
I think I should use the properties of Vieta's theorem then $x_1+x_2=-a$ and $x_1\cdot x_2=b$.
And I don't know how to solve it further.
The sum of two $U(0,\,1)$ IIDs has an Irwin–Hall distribution of PDF $1-|1-x|$ on $[0,\,2]$. Linearly transforming, we get the PDF of the sum of two $U(-1,\,1)$ IIDs, which here we take to be the $-x_i$: $a$ has PDF $\frac12(1-|a|)$ on $[-2,\,2]$.
We use the roots' distributions' symmetry to address the other coefficient. Since each $-\ln|x_i|\sim\operatorname{Exp}(1)$, $\ell:=-\ln|b|\sim\Gamma(2,\,1)$ has PDF $\ell\exp-\ell$ on $[0,\,\infty)$, i.e. $b$ has the even PDF $-\frac12\ln|b|$ on $[-1,\,1]$.