I'm studying probability. I came a cross "sampling from distributions". Given a probability density function $f_X(x)$, what I understood is that sampling means getting values of $x$ according to the distribution of probabilities over the $x$s. So that a value of $x$ (or an interval of $x$) that has a probability mass or density more than others will have a higher chance to appear in the sampling than others.
I have three questions here:
1- is my previous understanding correct?
2- I often see people sample from a normal distribution with zero mean and variance of 1. Why is that?
3- Does sampling from a normal distribution with zero mean and variance of 1 means that the chance of getting zeros or numbers around zero is higher than other numbers?