Interpreting a condition about CDF Let F(X) be a strictly increasing CDF which admits a positive density f(x).
Is the condition x/F(x) being non-increasing (aka, weakly decreasing) equivalent to saying that F(x) is convex?
If not, what is a somewhat intuitive meaning for this condition?
 A: They are equivalent in the sense that neither condition is ever true (for all $x$).  A CDF must satisfy $\lim_{x \to +\infty} F(x) = 1$ , so in particular $\lim_{x \to +\infty} x/F(x) = +\infty$ and it can't be non-increasing.
Similarly, if $F(x)$ is convex and $\lim_{x \to \infty} F(x) = 1$ then
$F(x) \ge 1$ for all $x$, and this can't happen for a CDF.
But if you're talking about these conditions being true on a bounded interval, they are not equivalent either.  You might note that $F(x)/x$ is the slope of the secant line from $(0,0)$ to the graph of the function at $(x,F(x))$.  It's not hard to sketch examples where on some interval
$x/F(x)$ is decreasing but $F$ is not convex, or where $F$ is convex but $x/F(x)$ is not decreasing.  In the picture below, for about $6.3 \le x \le 7.8$ we have $x/F(x)$ decreasing but $F$ not convex; for $3.2 \le x \le 4.2$ and we have $x/F(x)$ increasing but $F$ convex.

EDIT: Let's say $x > 0$.  $x/F(x)$ is decreasing for $x$ in some interval if in that interval the curve $y = F(x)$ crosses the secant line from $(0,0)$ to $(x,F(x))$ from below, as in the following picture (curve in red, secant line in blue):

It is increasing if in that interval the curve crosses the secant line from above, as in the following picture:

At a local maximum or minimum of $x/F(x)$ the tangent line to $y = F(x)$ passes through the origin.   You can have a local maximum where the curve is convex, or a local minimum where the curve is concave, but you can't have a local maximum where the curve is strictly concave or a local minimum where the curve is strictly convex. 
