# Beta distribution and normalization [duplicate]

Here in the 4 pictures in the last answer, is the vertical axe the probability? I.e. it seems to me that it is somewhat unnormalized : it has the value 2 in the 2nd picture and 3 in the 3rd picture. In the l.h.s. of each picture they write that this vertical axe really should be a probability. So why it is not in the range $$[0,1]$$? Also, they write

Note that as soon as you see your first Tail after the 3rd flip, the prior probability of p is now 0 at p=1 - ie there is SOME probability of seeing Tail.

So the value at each point should be in the range $$[0,1]$$, but it is not!

• Density is not probability Dec 22, 2018 at 12:21

These plots are PDFs, i.e. Probability Density Functions, which is used for dealing with continuous random variables. Specific values don't represent probabilities (e.g. $$f(x_0)\neq P(X=x_0)$$); instead they represent some kind of a measure of probability.
A final note: Instead of summing, you need to integrate the PDFs, in which you get $$1$$. Probability of a point is approximated as $$P(X=x_0) \approx f(x_0)dx$$, which is infinitely small, and need to be integrated for obtaining a proper sum.
• If PDF is $0$ at some point, then that point has no probability volume, using the approximation above. But, this doesn't mean that the PDF is probability, right? Dec 21, 2018 at 13:55
These 4 plots show the probability densitiy functions of some distributions, therefore they do not need to be in the range of $$[0,1]$$. They are not cumulative distribution functions.