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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 9 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 '14 at 19:47

Accepting the answer of @Glen_b (densities are not the probabilities) I would like to add more. I think it is also computational problems in both side software packages and how we use them. I used Mathematica 10 to illustrate, two methods are adopted to plot PDF of Weibull distribution with same parameters used in OP links. First method is just like to plot a function and second method is to proper defined distribution. The difference between two is plotted below

The code:

 f[\[Gamma]_] = \[Gamma] (x)^(\[Gamma] - 1) (x)^(\[Gamma] - 1)
f1[\[Gamma]_] = 
  ProbabilityDistribution[\[Gamma] (x)^(\[Gamma] - 1)
     Exp[-(x)^\[Gamma]], {x, 0, \[Infinity]}, 
   Assumptions -> \[Gamma] > 0];
Plot[{f[0.5], PDF[f1[0.5], x]}, {x, 0, 3}, 
 PlotStyle -> {Dotted, Dashed}, AxesLabel -> {"x", "f(x)"}, 
 PlotLegends -> {"Ordinary Method", "PDF Method"}, 
 PlotLabel -> "\[Gamma]=0.5"]

and plots

enter image description here enter image description here enter image description here enter image description here

Both functions are same but using them in different ways producing different probabilities. Same problem will occurs when we are using other package see for &\gamma=0.5& plot of OP link and Glen_b answer. For true values I think we must used PDF function while plotting density function in Mathematica.

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