Is there any literature discussing PDF after quantization? Consider a random variable with a known distribution. I want to know if we quantized the variable the how the probability density function(PDF) would change? Specifically if the original PDF is Rayleigh distributed what would be the distribution would look like after the quantization?
any discrete random variable has a p.d.f. that is a summation of dirac delta functions.

$$ p_\mathrm{y}(\alpha) = \sum\limits_i P_i \ \delta(\alpha - y_i) $$

where $\sum\limits_i P_i = 1 $.

if $y[n]$ is the quantization of $x[n]$:

$$ y[n] = \Delta \bigg\lfloor \frac{x[n]}{\Delta} \bigg\rfloor $$

where $ \lfloor \cdot \rfloor$ is the floor() function and $\Delta$ is the quantization step size.

then $y_i = i \Delta $ for some set of integers $i$.  the p.d.f. weighting constants become

$$ P_i = \int\limits_{i \Delta}^{(i+1)\Delta} p_\mathrm{x}(\alpha) \, d \alpha $$