I have a probability distribution for products entered a warehouse for example $f(x)$. I also have a probability distribution for products going out of the warehouse for example $g(x)$.
I want to know the probability distribution of inventory level (Inventory=Out-In)
Let the incoming items for an inventory period be random variable $X\in [0,\infty)$ with density measure $f$.
Let the outgoing items for an inventory period be random variable $Y\in [0,\infty)$ with density measure $g$.
Remark We will have to assume for the sake of modelling that the inventory is never depleted. In actual practice $Y$ should be restricted to no more than the current inventory level. Further we shall assume independence of incoming and outgoing items.
We wish to find the density measure $h$ of random variable $Z = X-Y$, the change in inventory level. So $Z\in (-\infty,\infty)$. (See the remark above.)
$$\begin{align}\because \forall z\ge 0, P(Z\le z \mid z>0) & = P(X\le z+Y) \\ & = \int_0^\infty P(X\leq z+y)g(y)\operatorname{d}y & \ni X\bot Y \\ &= \int_0^\infty \int_0^{z+y} f(x)g(y)\operatorname{d}x\operatorname{d}y \\[2ex]\therefore \forall z\ge 0, h(z) & = \frac{\operatorname{d}}{\operatorname{d}z} \int_0^\infty \int_0^{z+y} f(x)g(y)\operatorname{d}x\operatorname{d}y \\ & = \int_0^\infty f(z+y)g(y)\operatorname{d}y \\[2ex]\text{similarly:} \\ \forall z<0, h(z) & = \int_0^\infty f(x)g(x-z)\operatorname{d} x \\[3ex] h(z) & = \begin{cases}\int_0^\infty f(z+y)g(y)\operatorname{d}y & z\ge 0 \\ \int_0^\infty f(x)g(x-z)\operatorname{d} x & z\le 0 \end{cases} \end{align}$$