Chapter5 #31
Let U be a uniformly distributed random variable on [0,1]. What is the probability that the equation \(x^2 + 4Ux + 1 = 0\) has two distinct real roots x1 and x2?
In order for for the equation \(x^2 + 4Ux + 1 = 0\) to have two distinct real roots, the discriminant if the discriminant is larger than 0.
\(b^2 - 4ac\) > 0
\((4U)^2 - 4(1)(1) > 0\)
\(16U^2 - 4 > 0\)
\(16U^2 > 4\)
\(U^2 > \frac{1}{4}\)
U > \(\frac{1}{2}\)
Since U is uniform from [0,1] the probability is \(\frac{1}{2}\)