Assume $z$ is a standard normal variable. If $z<0$, then we accept it with probability 1. if $z\ge0$, we accept it with probability $e^{-mz}$, where $m>0$. I'm trying to figure out the pdf of this new r.v. $\hat{z}$ that is sampled using this procedure.
So far I've defined
$M^p = \int_{-\infty}^{0}z^p\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}dz + \int_{0}^{\infty}z^p\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}e^{-mz}dz$
$E(\hat{z}) = \frac{M_1}{M_0}$
$Var(\hat{z}) = \frac{M_2}{M_0} - (\frac{M_1}{M_0})^2$
This is essentially saying that the pdf is
$f(\hat{z}) = \frac{\begin{cases} \frac{e^{-\frac{\hat{z}^2}{2}}}{\sqrt{2\pi}},& \text{if } \hat{z} < 0\\ \frac{e^{-\frac{\hat{z}^2}{2}}}{\sqrt{2\pi}}e^{-m\hat{z}},& \text{if } \hat{z} \ge 0 \end{cases}}{M_0}$
Is what i'm doing correct? For what it's worth, sampling points using the described procedure and calculating the mean and the variance of the sample, they agree with the analytical ones.
Sort of. It looks like your math is correct, you're taking the total probability ($M_0$) and dividing each portion by that probability, normalizing it. I'm assuming you have to find the expectation and variance for other parts of your problem, and those look fine as well. The only thing I can see to look out for is that you say "accepting" and "rejecting" are the outcome states of your new random variable, $\hat{z}$, which leads me to think that the outcome states are $\{0,1\}$. If that's the case, then you have to write the pdf in terms of whether or not the outcome is indeed $0,1$, not the outcome of your original standard normal, $z$. This is probably easier, as you only have to take the probability that the result is accepted $(1/2 + \int_0^\infty\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}e^{-mz}dz)$, and the probability that the result is rejected, which looks similar but without the 1/2, normalize that if needed (I don't think you will have to though), and call it a day by setting the resulting bernoulli variable to have those probabilities.
As an aside, this is my first answer so sorry if formatting or conventions are not up to par.