Probability of a Function Having Complex RootsDate: 05/11/2000 at 18:28:08 From: Henrik Subject: Probability That a Function Has Complex Roots I found a similar problem in your archives, but note that this one is not quite the same: What is the probability, P, that the function f(x) = x^2 + px + q = 0 has complex roots? Consider the following cases: (a) 0 < p < 1 (b) 0 < p < 5 (c) 0 < p 0 < q < 1 0 < q < 5 0 < q Date: 05/12/2000 at 11:18:27 From: Doctor Anthony Subject: Re: Probability That a Function Has Complex Roots >What is the probability, P, that the function f(x) = x^2 +px + q = 0 >has complex roots? Consider the following cases: > >(a) 0 < p < 1 > 0 < q < 1 For complex roots you require B^2 < 4AC, that is: p^2 < 4q To find the probability of this, assuming that p and q are each uniformly distributed in the range 0 to 1, we set up rectangular axes with q along the horizontal axis and p on the vertical axis. The graph of p^2 = 4q is the boundary and we require a probability that we lie below this boundary. This will be given by the area under the boundary divided by the area of the 1x1 square (= 1). When p = 1, q = 1/4, so we integrate between 0 and 1/4 and then addthe rest of the area, which is (3/4) x 1. Area = INT(0 to 1/4)[p.dq] + 3/4 = INT(0 to 1/4)[2.q^(1/2).dq] + 3/4 = 2[(2/3)q^(3/2)] from 0 to 1/4 + 3/4 = (4/3)(1/4)^(3/2) + 3/4 = (4/3)(1/8) + 3/4 = 1/6 + 3/4 = 11/12 And so in this case the probability of complex roots is 11/12. >(b) 0 < p < 5 > 0 < q < 5 This time, when q = 5, p^2 = 20 and p = 4.47, so the curve remains inside the 5x5 box and we can simply integrate in the range 0 < q < 5. Area = INT(0 to 5)[p.dq] = INT(0 to 5)[2.q^(1/2).dq] = 2(2/3)q^(3/2) from 0 to 5 = (4/3)5^(3/2) (4/3)5^(3/2) and the probability is ------------ = 0.59628 25 So the probability of complex roots is 0.59628. >(c) 0 < p > 0 < q This time, Area below curve = (4/3)q^(3/2) Total area -> q^2 (4/3)q^(3/2) Probability = ------------ q^2 (4/3) = ------- -> 0 as q -> infinity. q^(1/2) So in this case the probability of complex roots tends to zero. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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