What is the name of a distribution within $[a, b]$ but with a sloped straight line density? What is the name of a distribution such that it looks like uniform distribution on $[a, b]$,
but the density is not a horizontal line, but a line with a certain slope.
 A: A distribution with a linear PDF can be considered a special case of a (truncated) Pareto distribution, Beta distribution, or power distribution.  Only particular values of the parameters in these distribution families will give a linear PDF, of course.

Among other things, such a distribution is a truncated Generalized Pareto Distribution.  The Wikipedia parameterization of the PDF (for the case where the endpoints $a$ and $b$ are finite, as they must be for a linear graph) can be expressed in terms of the basic function
$$f(x; \eta) = (1 - \eta x)^{1/\eta - 1}$$
for $0\le x \le 1/\eta$.  (This can then be rescaled by $\sigma\ne 0$ and shifted by $\mu$ to obtain the most general form.  Here I have set $\eta = -\xi$ which will be a positive number.)
Evidently this function is linear in $x$ if and only if $1/\eta - 1=1$; that is, $\eta = 1/2$ (and so $\xi=-1/2$).  Its graph equals zero at the endpoint $x=1/\eta = 2$.  By truncating it, though, we would obtain the general form stated in the question.

The same thing can be obtained by truncating a generalized Beta Distribution.  Its PDF is proportional to
$$x^{\alpha-1}(1-x)^{\beta-1}$$
for $0\le x \le 1$, whence the Beta$(2,1)$ and Beta$(1,2)$ distributions are linear.  As with the Pareto$(\xi=-1/2)$ distribution, the graphs of these PDFs are zero at one endpoint.  The general linear PDF described in question is obtained in the same way via truncation, rescaling, and shifting.

Finally, Mathematica defines a "power distribution" as one having a PDF proportional to 
$$f(x; k, a) = x^{a-1}$$
for $0 \le x \le 1/k$.  The case $a=2$ gives a linear PDF, identical to Beta$(2,1)$.  Rescaling it by $k$ and recentering it with a parameter $\mu$, and (once again) truncating it will yield the general PDF described in the question.
A: It might be the triangular distribution. The general triangular distribution with support $[a,b]$ has 1 parameter $c\in[a,b]$ corresponding to the mode. When $c=a$ or $c=b$, the p.d.f. is a straight line segment.
