In this paper, we seek to find out the probability of obtaining real roots of a quadratic equation AX2+BX+C=0, with A≠0, when the coefficients are independent, identically distributed uniform variates. The exact value of the roots can be obtained from the coefficients and the discriminant indicates if the roots are real or imaginary. Here, we consider the uniform distribution U(-θ,θ) and find the probability of obtaining a real root to be 62.7%. This is done through simplification of the problem, analysis of the probability distribution of B2 for both U(-1,1) and U(0,1), and final evaluation using conditional probability. Calculations are simplified by the fact that B2≥4AC is always true when AC≤0. We leverage on the fact that the probability of obtaining real roots when coefficients are sampled from U(0,θ) is 25.4%. We verify the result experimentally through Monte Carlo simulation and present the desired supporting data accordingly.
 Nick Peterson (https://math.stackexchange.com/users/81839/nick-peterson), Probability that a quadratic equation has real roots, URL (version: 2020-09-08): https://math.stackexchange.com/q/3819057. The theoretical analysis is an expanded version of this answer. [accessed 9th Sept., 2020].
 Rozanov, Y. A. (1977). Probability Theory: A Concise Course, Dover Publications, Inc., N.Y., Dover Edition.
 D’Aurizio, J. (https://math.stackexchange.com/users/44121/jack-daurizio), Probability of a real root when coefficients are sampled from U(0,1) is ≈0.2544134190: Probability that a quadratic equation with random coefficients has real roots, URL (version: 2016-01-10): https://math.stackexchange.com/q/1607406. [accessed 9th Sept., 2020].
 Felix Marin (https://math.stackexchange.com/users/85343/felix-marin), Probability that a quadratic equation has real roots, URL (version: 2020-09-09): https://math.stackexchange.com/q/3819074 - An interesting alternate solution using Iverson brackets (and integration) [accessed 9th Sept., 2020].
 Gentle, J. E. (2009). Computational Statistics, Springer-Verlag, N.Y., Ist Ed.
How to cite this paper
On the Probability of Real Roots in a Quadratic Equation with Coefficients as i.i.d U(-θ,θ) Variates
How to cite this paper: Subhomoy Haldar, Soubhik Chakraborty. (2021) On the Probability of Real Roots in a Quadratic Equation with Coefficients as i.i.d U(-θ,θ) Variates. Journal of Applied Mathematics and Computation, 5(1), 48-55.