I have a question that asks to suppose that X and Y are two independent random variables both uniformly distributed on the unit interval [0,1]. I am asked to calculate the probability that
(a) $4X \leq Y^2 +2 $
(b) $4Y \leq Y^2 +2 $
(c) $2(X+Y) \leq Y^2 +2 $
I am not looking for the solution to this problem but would be thankful if somebody could help explain the right way to get started with these types of problems, because I am having difficulty with it.
This is a good question. This is not something that is easy to comprehend when you first encounter it.
These are all problems in conditional probability.
In (a) the sides of the inequality are independent, so you can just work out the density of $Z=Y^2+2$ and integrate it against the probability it will be greater than $4X$, which is just $z/4$ floored by zero and ceilinged by one.
In (b) the sides are completely dependent and this is really an algebra problem rather than a probability problem. Then once you figure out the values of $Y$ that make it true, you can see the probability that a uniform random variable on $[0,1]$ realizes those values. (If the solution set is an interval, it's just the length of the intersection of that interval with $[0,1]$.)
In (c) the two sides are dependent but not totally dependent. The easiest thing to do is to restate the inequality to have sides that are indepedent and then use the same method as for (a).
Hope this helps.