Polynomials and Probabilities

Question

For a second degree polynomial (i.e. with three terms) with coefficients chosen randomly over the interval [-10,10], what is the probability of two real roots?

Extend your analysis to a similar third degree polynomial. Now, what is the probability of only a single real root?

Extend your analysis even further to a similar arbitrary odd-degree polynomial. How does the probability of a single real root scale with the degree of the polynomial?

Answer 1

Well here is some LOOSE logic and food for thought, just to get you started hopefully in the right direction.

I apologize for the lack of mathematical rigor.

as you know b^2 - 4ac > 0 in order for the roots to be real.

so b^2 > 4ac

where a,b,and c are uniformly continuous random variables in the range [-10,10]

b^2 will always return a positive value, but NOT always positive for 4ac .In fact b^2 must be in [0,100]

So on the square of [-10,10] for each side, 4ac will be negative when b is in [-10,0] and c is in [0,10] or vice versa.

That puts the probability at least 1/2.

Now with the factor of 4 in play, in order for b^2 to have a shot at pulling it off, AC must be AT MOST 25,

since b^2 is AT MOST 100.

for the regions where 4ac is positive, namely b and c in [0,10] or b and c in [-10,0], their product must be at most 25.

so the slices we are interested in is [2.5,10] or [-10,-2,5] for b and c.

So the rest of the probability boils down to the product of two random numbers in the domain [2.5,10] being 25 or less (and then of course doubling it to account for the negative)

Upon a little bit of research, the pdf of the product of two uniform vars is -log z where z = ac in this case.

So if you can derive a decent pdf out all of this, you can calculate that probability plus 1/2.

Cheers