Suppose that $X\sim N(0,1)$. Find the distribution of $Y$ if $Y=X\mid X>0$.
Normally when I find conditional distributions, it's when one random variable is conditioned on another, and is the quotient of a joint and marginal PDF. However, in this situation, this is a random variable that's conditioned on its own value... so I'm not quite sure what steps I should take to derive the PDF. How do you do this?
Remember, I'm interested in HOW to derive the PDF, not simply knowing what it is.
As per usual, it is probably helpful to derive the CDF and then obtain the PDF by differentiation. We have $$ P(X>x \mid X>0) = \frac{1}{P(X>0)}P(X> x,X>0) = \frac{P(X>x)}{P(X>0)}1_{x>0} + 1_{x\le 0}.$$ The conditional CDF is just one minus this, so the PDF is $$ f(x\mid X>0) = -\frac{d}{dx}P(X>x \mid X>0) = -1_{x>0}\frac{1}{P(X>0)}\frac{d}{dx}P(X>x) = \frac{1_{x>0}}{P(X>0)}f_X(x).$$
In retrospect it might have been easier to simply guess that the conditional PDF should be the same as the old one for positive numbers (since the relative proportions amongst the positive numbers must be the same), and just needs a new normalization factor so it integrates to one over the positive numbers instead of over the real line.