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I am looking at distributions here. I am confused that some PDFs (e.g. exponential, Weibull with gamma = 1) have a P(X) = 1 about the axis. I thought integrating over the entire distribution should yield 1. But these distributions obviously equal more than one over their area. What am I misunderstanding?

Thank you

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up vote 5 down vote accepted

When you say "obviously" --- you're mistaken. In fact, those (pretty clearly) all have area-under-the-curve close to 1.

I've drawn the same four densities, and on top of the image I've added some shapes in red (each composed only of triangles and rectangles for which the computation of area is simple) - each one has area under the red "curve" equal to 1.

enter image description here

Hopefully you can now see that the areas under the curves must actually be quite close to 1.

Algebraically, of course, they all demonstrably have area under the curve equal to 1.

(You may have been partly confused by the height of the pdf exceeding one, but that would only be a problem if density was probability, which it isn't.)

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Thank you very much –  horse hair May 25 at 19:47

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