Find the probability that the roots of the equation h(t)=0 are real where h(t)=t^2+2tx+y

Question

assume that x and y are independent with x ~EXP(2) and Y~Exp(1) Find the probability that the roots of the equation h(t)=0 are real where h(t)=t^2+2tx+y

 

Could you please help me how can I find the probability?

Answer 1

I am not sure what you mean by finding the probability, because we can know with certainty that the roots are real if the discriminant D ≥ 0.

 

D = (2x)² - 4(1)y = 4x² - 4y = 4(x² - y)

 

So as long as x² - y ≥ 0, the roots are real, with certainty.

 

If x = e² and y = e¹ = e, then

 

x² - y = (e²)² - e = e⁴ - e > 0 always, so D > 0 always under these conditions, and the roots are real.