If the random variable Q has a continous uniform distributio

If the random variable Q has a continous uniform distribution with parameters A=0 and B=3, find the probability that the roots of the equation g(t)=0 are real where g(t)=4t^2+4Qt+Q+2

Solution

The quadratic equation has real roots if discriminant >0

i.e. 4Q²-16(Q+2)>=0

(q-2)(q+1) >=0

I.E. Q <-1 or Q >2

But Q is uniform with a =0 and b =3

Hence prob = Prob Q>2 = 1/3