# Laplace Mechanism Proof: Why this product operator?

The equation below shows the proof of Laplace mechanism for differential privacy. I am not understanding the product operator, is this a common rule?

$$\frac{p_x(z)}{p_y(z)} = \prod_{i=1}^{k}\left(\frac{exp(-\frac{\varepsilon |f(x)_i - z_i|}{\Delta})}{exp(-\frac{\varepsilon |f(y)_i - z_i|}{\Delta})}\right)$$

While $$p_x$$ and $$p_y$$ denote the pdf of mechanisms $$\mathcal{M}_L(x,f,\varepsilon)$$ and $$\mathcal{M}_L(y,f,\varepsilon)$$ respectively. Important to notice that $$x$$ and $$y$$ are neighbor datasets. $$x, y, z \in \mathbb{R}^k$$

The definition of Laplace Mechanism: $$\mathcal{M}_L(x, f(\cdot), \varepsilon) = f(x) + (Y_1,\ldots,Y_k)$$ Once each $$Y_i \sim Lap(\Delta f/\varepsilon)$$ they are indepedent of each other. So we can calcule the joint probability of them using the product of marginals.
As we know $$p_x = \mathcal{M}_L(x,f,\varepsilon)$$ and $$z \in \mathbb{R}^k$$, thus is possible to do:
$$p_x(z) = \frac{\varepsilon}{2\Delta f} exp\left(-\frac{\varepsilon |f(x) - z|}{\Delta f}\right) = \prod_{i=1}^k \frac{\varepsilon}{2\Delta f} exp\left(-\frac{\varepsilon |f(x)_i - z_i|}{\Delta f}\right)$$