Question

IfYis a uniformly distributed random variable over (0,5), what isthe probability that the roots of the equation 4x2+ 4xY+Y+ 2 = 0 are both real?

Answer 1

The given quadratic has real roots if its discriminant is non-negative, so the probability is

`P(16Y^2-16(Y+2)\ge0)=P(16Y^2-16Y-32\ge0)`

`=P(Y^2-Y-2\ge0)`

Complete the square:

`Y^2-Y-2=Y^2-Y+\frac14-\frac94=\left(Y-\frac12\right)^2-\frac94`

`\implies P(Y^2-Y-2\ge0)=P\left(\left(Y-\frac12\right)^2-\frac94\ge0\right)`

`=P\left(\left(Y-\frac12\right)^2\ge\frac94\right)`

`=P\left(\left|Y-\frac12\right|\ge\frac32\right)`

`=P\left(Y-\frac12\ge\frac32\text{ and }Y-\frac12\ge-\frac32\right)`

`=P\left(Y\ge2\text{ and }Y\ge-1\right)`

`=P\left(Y\ge2\right)`

`=\displaystyle\int_2^\infty f_Y(y)\,\mathrm dy`

where
`f_Y(y)` is the PDF of
`Y`,

`f_Y(y)=\begin{cases}\frac15&\text{for }0\le y\le5\\0&\text{otherwise}\end{cases}`

`\implies P(Y\ge2)=\displaystyle\int_2^5\frac{\mathrm dy}5=\boxed{\frac35}`