Why Reparameterization Trick does not work with discrete latent variables? I came to know from the Youtube Video here (Timestamp 1:03:55) that Reparameterization trick only works for continuous latent variable. But, I am not clear as to why it does not work for discrete latent variables. Reparameterization changes z as $\mu$+$\epsilon$*$\sigma$. We can determine mean and standard deviation of discrete variables too. So, why does the Reparameterization trick not work with them?
Could anyone please provide me insight on this?

The re-parameterization $x=\mu + \epsilon \sigma$ where $\mu, \sigma>0$ are the outputs of a neural network and $\epsilon \sim \mathcal N(0,1^2)$ is a standard normal variable implies that $x\sim \mathcal N(\mu,\sigma^2)$, i.e. $x$ has normal distribution with mean $\mu$ and standard deviation $\sigma$. Therefore, $x$ is not a discrete random variable.
If you have a different reparameterization in mind, or a different sampling of $\epsilon$, then that would change this result, but it's not clear if that's what you're asking.
