Let denote a random variable uniformly distributed over . Compute the conditional distribution of given that
where
So if I understand this correctly, we desire
How do I proceed from here, assuming I'm correct?
Let denote a random variable uniformly distributed over . Compute the conditional distribution of given that
where
So if I understand this correctly, we desire
How do I proceed from here, assuming I'm correct?
Hello,
You're applying the formula for the conditional density in a bizarre way. It's the conditional density of a random variable with respect to another random variable, not with respect to an event. And here, {U>a} is an event. You can't write that. It makes no sense to take the "density" of U>a !
U>a should remind you of the cumulative density function. And it's like a density because it determines the distribution of a random variable. And since it works with probabilities, it will be better
So we're looking for P(U<x|U>a). Of course, the value will depend on whether x>a or not, and if it's in (0,1). But it's not an insurmountable obstacle. And if you want the pdf, you know what to do once you get the conditional cdf