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I know the integral represents the probability, but what does the height represent? It can’t be the number of times that a feature appears, because the function is continuous.

I understand that it is the "probability density", but I don't really understand what is meant by that.

JobHunter69
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    Appears to be a duplicate of http://stats.stackexchange.com/questions/86094/what-is-a-density-function ... also see this answer: http://stats.stackexchange.com/questions/85436/what-could-it-mean-to-rotate-a-distribution/85447#85447 and this one: http://stats.stackexchange.com/questions/4220/can-a-probability-distribution-value-exceeding-1-be-ok/4223#4223 which add some additional insight – Glen_b Jun 20 '16 at 07:05

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You are correct to say it represents the value of the probability density - though this is obviously a kind of tautology.

The probability density is effectively the probability that would be achieved if the height was constant and the function was integrated across one unit on the x-axis. Hence you can use it to understand the relative probability at different points on the graph.

However, as it's a continuous function it has a value of zero at any given point (i.e. having a width of zero), and only becomes non-zero when integrated between two points.

Robert de Graaf
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  • By claiming the density is a "probability that would be achieved" you open the reader up to the misconception that a density must be $1$ or less. You also rule out many commonly used densities by implicitly assuming they are "continuous functions": that's clearly not the case for some Beta distributions, all Exponential distributions, and many more. – whuber Jun 20 '16 at 14:29