I understand probability as the likelihood of something happening. I understand that a probability density function shows the probability that a random variable takes a range of values. But what exactly is the y axis of a pdf? I keep on seeing the phrase “probability per unit length” or equivalent, but what does the “per unit length” mean exactly?
edit: thank you all for your responses.
In: 3
Probability density is a mathematical tool to help calculate actual probabilities.
The y-axis is the thing it needs to be for the probability density function to work. It doesn’t really have meaning on its own.
The “probability per unit length” thing comes from working backwards – as does probability density itself.
The area under a probability density function gives us a probability. Area has dimensions of length * height, so if:
> length * height = probability
> height = probability / length
So in this case “length” is whatever your x-axis is.
It’s basically probability per some section of the graph. So in practice, let’s say you had probability density function that represented people’s heights. Let’s you want to find the odds that a person is 180 cm tall. In practice, the odds of that are practically 0, because you would need to be 180 to a microscopic degree. You can’t be 180.0000000001 or 179.999999999. So you normally don’t work with exact values, you work with ranges. So let’s say you wanted to know the odds that someone is between 179.5 cm tall and 180.5. Height would be the X axis, and if you take that section of the probability density function between 179.5 and 180.5 and calculated the area, that would be the odds that your height is between 179.5 and 180.5.
A probability density function is an infinite regression of a count (y) to category (x) relationship, the categories being ranges of continuous values reduced so the actual range covered approaches zero (the regression), at which point the actual count becomes irrelevant.
The key idea is that the total area under the PDF needs to be 1. This follows from one of the main axioms of probability, i.e. that the probability of *something* happening is 1. The actual units of the y-axis are scaled to make this true.
As an example, consider a random variable that can take on any value between 0 and 10 with equal probability. The corresponding PDF looks like a rectangle that runs from x = 0 to x = 10 with some height. You need the area of that rectangle (area under the PDF) to be 1, hence the height must be 0.1.
What’s density? Mass per unit length/area/volume. If you have a string with mass (that is thin is enough that you can treat it as 1D), then the mass density of the string is mass per unit length.
If you want to be more precise about it, then this is an issue that was actually challenging to Newton/Leibniz. They were not able to adequately explain the notion, and even philosophers from back then know there was something fishy about their explanation. Basically, they said that mass per unit length is just total mass divided by total length where the length is infinitesimal, that is, smaller than all positive length, but bigger than 0. Of course, the idea of a length bigger than 0 but smaller than all positive number is a kinda sus.
Luckily, later on, we were able to remedy it. This is due to a technical tool, called the Lebesgue differentiation theorem. You can define the probability density at a point to be the limit of the ratio of total probability on an interval over the length of that interval, as the interval’s length approach 0. Note that this limit is not guaranteed to exist. But the points where the limit does not exist form a set so small that it has no impacts whatsoever on the final probability.
Thus, you should be aware that the probability density might not be definable point-wise, but *a* probability density function still exist.
Latest Answers