While going through a paper on Random Sequential Adsorption ("Dynamics of polydisperse irreversible adsorption: a pharmacological example" by Erban et. al, 2007, https://www.worldscientific.com/doi/abs/10.1142/S0218202507002091), I come across on page 5:
"...the probability (per unit time) of adsorbing the polymer segment of length z $\leq x = x_2 - x_1$ as
$$ \xi(z, w - x_1, x_2 - w)= \begin{cases} \frac{2(w - x_1)}{z}, &\text{for } w \in [x_1, x_1 + \frac{x}{2}]; \\ 1, \quad &\text{for } w \in [x_1 + \frac{z}{2}, x_2 - \frac{x}{2} - \frac{z}{2}]; \\ \frac{2(x_2 - w)}{z}, &\text{for } w \in [x_2 - \frac{z}{2}, x_2]; \\ \end{cases} $$ and the probability of adsorbing the polymer segment of length $z > x$ as $$ \xi(z, w - x_1, x_2 - w)= \begin{cases} \frac{2(w - x_1)}{z}, &\text{for } w \in [x_1, \frac{x_1 + x_2}{2}]; \\ \frac{2(x_2 - w)}{z}, &\text{for } w \in [\frac{x_1 + x_2}{2}, x_2]." \end{cases} $$
Now, in these formulas, the length of the domain is $[0, 1]$, the small interval is of length $z$ with midpoint $w$ and the new gap is [$x_1$, $x_2$]. Later on, $\xi(\cdot)$ is referred to as the "probability density function".
My question is: Is $\xi(\cdot)$ a probability density function? If I integrate over the whole domain, I would expect to get a result of $1$ if so. But in fact, I get a result of $x_2 - x_1 + \frac{z}{2}$ for the upper formula and $\frac{(x_2 - x_1)^2}{2z}$ for the bottom formula. Should this not integrate to $1$?
