Let A, B, and C be independent random variables, and consider the random-coefficient quadratic equation
What is the probability that the equation has real roots when A, B, and C take on the value 1 with probability and -1 with probability ?
What is the probability that the equation has real roots when A, B, and C a have a distribution?
As Mr Fantastic said, for part b of this question I need to find the probability of the determinant when it is greater than zero. Alternatively I could say that I require . But I am getting confused with how to do this if A,B, and C all have a uniform distribution.
I know the pdf and the distribution function for a uniform distribution, but is this a case of conditional probability? If so, how do I apply it here. Continuous disrtibutions confuse me
If instead A,B,C ~ uniform(0,1) and are still independent, one way to do it is to compute the distribution of the random variable B^2 - 4AC. The chance of real roots is then the probability that this quantity is positive. I recall that someone on Math Help Forum mentioned recently the distribution of the product of standard uniforms (which you need for the 4AC part), but then you have to find the distribution of the square, and finally the difference too. The distribution would seem to be prety messy. I'm not sure whether folks on this list would deign to use Monte Carlo simulation or numerical methods to do such a calculation, but I found this probability to be about 0.254 using RAMAS Risk Calc. You can also do it with Monte Carlo simulation in R (The R Project for Statistical Computing) with a
simple script like
many = 2000000
A = runif(many)
B = runif(many)
C = runif(many)
arg = B^2 - 4*A*C
length(arg[0<arg])/many
Is resorting to such approaches bad form?
Not at all.
However, you can find the result analytically. The probability that the roots are real is the same as the probability the , and
..... by independence
..... since f is the Uniform p.d.f.
where the region of integration in all the triple integrals is the region where , , , and .
If you can work out the integral (i.e., find the volume of the region), then you are done.