How to calculate the CDF of symmetric random variable over the Y axis, given that random variable is greater than 0?
Given $Z\sim\operatorname N(0,1),$ find $P(z<Z\mid Z>0)$
I tought of making a condition within the given condition. But $P(z<0)$ which I think shall be involved in the derived function is not something I understand how to write properly when again I write a condition inside a condition. I'd (mistakengly) say that $P(z<0)=0.5$ but this is 100% wrong. small $z$ is not Random Variable, it's just a variable. Could you help me calculate this and write down properly (preferably with condition inside a condition, so it will align with how I see this?)
EDIT: I was thinking of writing down something like this - $P(z<Z|Z>0)=P(z<0,z<Z|Z>0)P(z<0)+P(z>0,z<Z|Z>0)P(z>Z)$
EDIT2: Could you help me please particulary with $P(z>0,z<Z|Z>0)$? How to calculate it?
EDIT3: I understand z is just a variable, but there's for sure a proper way to write this using the same idea of Law of Total Probability.
Use the standard formula for conditional probability. You're trying to calculate$$\frac{P(z<Z\land Z>0)}{P(Z>0)}=\frac{P(Z>\max(z,\,0))}{1/2}=2\Phi(-\max(z,\,0)),$$where $\Phi$ is the $N(0,\,1)$ CDF. For $z\le0$, the result is $2\Phi(0)=1$; for $z>0$, the result is $2\Phi(-z)$.