probability density function tail off I am reading a paper on EKF SLAM, and I came across this sentence on proability distribution functions that wasn't very clear to me:

The depth coordinate of such features has a probability density that rises sharply at a well-defined minimum depth to a peak, but then, tails off very slowly toward infinity—from low parallax measurements, it is very difficult to tell whether a feature has a depth of 10 units rather than 100, 1000, or more.

Here features are the features in an image and depth is it's distance from the sensor. I was wondering how this pdf would look like - something like a sigmoid? Any pointers on this would be very helpful!
Paper: Inverse depth parametrization
 A: A sigmoid can be a cumulative probability distribution function but is not a probability density function, since it approaches a positive constant at infinity, so the area under it is infinite.
The derivative of a sigmoid goes to $0$ at $\pm\infty$ more slowly than does a normal desnity, but still may be faster than what they have in mind.
Possibly a Cauchy density was intended, or a Student's t with a low number of degrees of freedom.  The standard Cauchy distribution is
$$
f(x)\,dx = \frac{dx/\pi}{1+x^2}.
$$
This satisfies
$$
\int_{-\infty}^\infty |x| f(x)\,dx = \infty,
$$
so there is no expected value, let alone a finite variance.  Its first, second, and third quartiles are $-1$, $0$, and $1$ respectively.  The distributions
$$
f\left( \frac {x-m} s \right)\frac{dx} s
$$
are the Cauchy distributions.  They are a location-scale family with location parameter $m$ and scale parameter $s$.
However, you've given us so little information that it is impossible to be sure what distributions the author had in mind.
