What is the probability that both roots of the equation $Ax^2 + Bx + C = 0$ are real?

Question

Given this problem as part of prep for a test. We've done the same problem without A being a random variable, but I am completely stumped as to how to accomplish this one with three r.v.s

I know the joint is $1/288$ and that $B^2>4AC$ but cannot convert this to a happy integral.

Let $A$, $B$, and $C$ be independent random variables, uniformly distributed over $[0,4], [0,8]$, and $[0,9]$ respectively. What is the probability that both roots of the equation $Ax^2 + Bx + C = 0$ are real?

Thanks,

Just to make some connections: 29242 is the node for the basic version with all three uniforms being in $[0,1]$. There are similar questions where the ranges of the unifs differ: 3182777, 1528677, and 3545048. — Lee David Chung Lin, Feb 13 '20 at 09:21

score 1 · Answer 1 · answered Jul 24 '14 at 23:22

1

If you've done it for fixed $A$ then do the same thing except replace your $p_C$ distribution with

$$ p_{AC}(z)=\frac{1}{|Z|}\int_Z p_A(x)p_C(z/x) dx $$

answered Jul 24 '14 at 23:22

lemon

3,548

What is the probability that both roots of the equation $Ax^2 + Bx + C = 0$ are real?

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