what does $+$ mean in $f(x) = (1 - |x|)_{+}$? I'm doing homework and in one of the problem a density function is given by $f(x) = (1-|x|)_{+}$. I don't know if the $+$ is a mistypo or not. Anyone familiar with this? 
If it matters, $Y$ is a random variable with density $f(x) = (1-|x|)_{+}$ and I'm supposed to calculate the characteristic function of $Y$
 A: It is sometimes useful to break a function -- or a real number -- into its "positive part" and its "negative part". The parts are defined by the functions
\begin{align*}
|x|_+ &:= \begin{cases} x &\text{if } x \ge 0 \\ 0 &\text{if } x \le 0\end{cases} \\
|x|_- &:= \begin{cases} 0 &\text{if } x \ge 0 \\ -x &\text{if } x \le 0\end{cases}
\end{align*}
The defining property of these functions is that, for any real number $x$,
$$
x = |x|_+ - |x|_-
$$
and moreover, either $|x|_+ = 0$ or $|x|_- = 0$, or both. Equivalently,
$$
|x|_+ \cdot |x|_- = 0.
$$
Another nice fact that may explain the notation using absolute values bars is
$$
|x| = |x|_+ + |x|_-.
$$
But what do these two functions, $|x|_+$ and $|x|_-$ (postive and negative part) mean? It may be helpful to look at these pictures, which I got from this online article about it.
Here is a function $f(x)$:

and here is it broken down, the two functions $|f(x)|_+$ and $|f(x)|_-$:

Maybe you can more or less get a good idea of positive and negative part directly, just looking at the images.
