What is a linear probability density function? In the following question, what is meant by linear probability density function? Is it a uniformly distributed variable or triangularly distributed? Thanks in advance.

The kinetic energy of any object in motion is given by the $E(v)=\frac{1}{2}mv^2$, where $v$ is the velocity in m/s. Someone measures the speed of students walking along Symonds St. to the Engineering buildings, and notices that the speed can be described by a linear probability density function in the range 0-1m/s.
(a) Write an equation for the probability distribution and sketch the probability density function. Make sure to label values on both axes.

 A: To me, it seems like it means the probability density, $\sigma(v)$, a function from the interval $[0,1]$ to the reals, is a linear function. So you simply have $\sigma=a v+b$ (linearity), $\int_{[0,1]}\sigma dv=1$ (real probability distribution), and $\sigma\ge 0$ (real probability distribution). You can use these conditions to eliminate one of $a$ or $b$ and put constraints on their magnitudes to ensure sigma is always positive.
A: If the speed is random but constrained to 0-1 m/s, you have to guess what is meant.
I'm guessing it is any distribution with linear shape, and of course area 1.
It could go from 0 to 2, from 2 to 0, from 0.5 to 1.5, or simply be the uniform distribution from 1 to 1. In any case, it's got a Y-intercept (the pdf of speed 0) and a slope.
If it has a non-zero pdf for speed 0, that means it's possible to see some students walking at a speed near zero, so they might take a long time to travel one meter.
That leads me to suspect the distribution wanted is the one with intercept 0 and slope 2,
but really it is not a well-worded question.
A: In this case, linear refers to the axis for speed being linear (not exponential or logarithmic). A probability density function is described as the normal distribution of an event. See the wiki page.
