Probability of triangle inequality given probability density functions For three random variables $A$,$B$,$C$, I have their probability density functions $f_A$, $f_B$, and $f_C$. The PDFs are polynomial.
I am trying to calculate the probability that $a+b>c$, or $$Pr[a+b>c]$$ using the PDFs. Is there a method for this?
I know that I can calculate things like $Pr[x_0 \leq a \leq x_1 ] = \int_{x_0}^{x_1}f_A(x) \ dx$.  but I'm not sure how to determine something like $Pr[a+b>c]$.


I know that I can calculate things like $\displaystyle\Pr[x_0≤A≤x_1]=\int_{x_0}^{x_1}f_A(x)\,\mathrm d x$. but I'm not sure how to determine something like $\Pr[A+B>C]$.

Basically, it is the same principle. You integrate over the supported domain where the condition is also met; you just do so for all three variables.
$\qquad\begin{align}\Pr(A+B>C) &= \iiint_{x+y>z} f_{A,B,C}(x,y,z)\,\mathrm d z\,\mathrm d y\,\mathrm d x\\[1ex]&=\int_{-\infty}^\infty\int_{-\infty}^{\infty}\int_{-\infty}^{x+y} f_A(x)\,f_B(y)\,f_C(z)\,\mathrm d z\,\mathrm d y\,\mathrm d x\end{align}$
If the variables are independent, then $f_{A,B,C}(x,y,z)=f_A(x)\,f_B(y)\,f_C(z)$, but if they are not, then you will need to know what is that joint probability density function.
