# 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.

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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.

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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.

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In other words, the probability distribution is some linear function: $ax+b$. The condition that the total area under this line, between 0 and 1, is 1 (so that the total probability is 1), gives you: $a/2 + b = 1$ but without more information we can't constrain these variables further --- in other words, we've got two unknowns, $a$ and $b$, and only have one piece of information (that the total probability is 1). You might be expected to set $a = 0$, although there's no reason for that. – gj255 Apr 13 '14 at 19:07
@gj255: What you would do is take your sample of n velocities and sort them to get an empirical distribution. Then see which $av+b$ line fits it best. – Mike Dunlavey Apr 14 '14 at 0:49

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.

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Thank you for the reply. Can you please clarify/explain how you know that the probability distribution is defined as the normal distribution? – Ganan Apr 13 '14 at 4:44
@Ganan With the example above, it is unlikely that enough data points were collected to state that the probability density function is a normal distribution. It is likely that the distribution is a normal distribution since the speed of the walkers have a continuous range and that the average speed would be 0.5 m/s. Also, since the question did not specify the data was skewed, it is easier to assume the data has a normal distribution. – LDC3 Apr 13 '14 at 5:00