Let the continuous random variables $X,Y$ be independent of each other and uniformly distributed on $[0,1] $ i.e. $X,Y \sim U[0,1]$, what is the probability that $t^2+Xt+Y=0 $ has a real root?
I am trying to solve it with Geometric probability models, knowing $X^2-4Y \ge 0$. But I think the real answer should be related with r.v. and the properties of uniform distribution. What is the right way to solve the problem?
Anybody could help?
Many Thanks.
As you said, the probability to have a real root is $P(X^2-4Y\geq 0)$
you can calculate it integrating
$$\int \int_{x^2\geq 4y} f(x,y)dxdy$$
being $f(x,y)=1$ the double integral is equivalent to the integration area
$$P(X^2\geq 4Y)=\int_0^1\frac{x^2}{4}dx=\frac{1}{12}$$