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
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.
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