The three random variables, a, b and c are independent and each uniformly distributed in theinterval (0, 1].Find the probability that the equation ax? + bx + c = 0 has 2 real roots.

Answered: The three random variables, a, b and c…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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The three random variables, a, b and c are independent and each uniformly distributed in the
interval (0, 1].
Find the probability that the equation ax? + bx + c = 0 has 2 real roots.

Expert Solution

Step 1

We can find the roots of any quadratic equation $a x^{2} + b x + c = 0$ using the discriminant as follows:

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Let $y = b^{2}$ .

The distribution of y is given by

$f (y) = \{\begin{cases} \frac{1}{2 \sqrt{y}}, i f 0 \leq y \leq 1 \\ 0, o t h e r w i s e \end{cases}$

Let $z = 4 a c$ . Then the distribution of z is given by

$f (z) = \{\begin{cases} \frac{- \log (\frac{z}{4})}{4}, i f 0 \leq z \leq 4 \\ 0, o t h e r w i s e \end{cases}$

Since y and z are independent, the joint pdf is given by

$f (y, z) = \{\begin{cases} \frac{- \log (\frac{z}{4})}{8 \sqrt{y}}, i f 0 \leq y \leq 1 a n d 0 \leq z \leq 4 \\ 0, o t h e r w i s e \end{cases}$

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The three random variables, a, b and c are independent and each uniformly distributed in the interval (0, 1]. Find the probability that the equation ax? + bx +c = 0 has 2 real roots.