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
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.
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