Two Uniform Independent Random Variables: When is one greater?

Question

You have two independent random variables: $X$ and $Y$, which are both uniformly distributed over $(0,1)$.

Consider the inequality $X^2- 4Y < 0$. What percentage of the time is the inequality true?

Answer 1

We want the probability that $Y\gt \frac{X^2}{4}$.

Draw the square with corners $(0,0)$, $(1,0)$, $(1,1)$, and $(0,1)$ where the joint density function "lives." Draw the parabola $y=\frac{x^2}{4}$. We want the probability that $(X,Y)$ ends up in the part $P$ of the square which is above the parabola.

Since the joint density is $1$, this probability is the area of $P$. The picture shows that this area is $$\int_0^1 \left(1-\frac{x^2}{4}\right)\,dx.$$ It is somewhat more pleasant to find instead the area of the rest of the square. This is $$\int_0^1 \frac{x^2}{4}\,dx,$$ which is $\frac{1}{12}$. Thus the required probability is $\frac{11}{12}$.