08 Feb '16 11:12>7 edits
Originally posted by twhiteheadOh hell, please lets no go there. That looks to me like a conceptual nightmare guaranteed to cause complex confusions.
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Even more fun would be to let all irrational numbers be twice as likely as rationals. I wonder if a probability density function can be defined in that case.
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The concept of just the 'ordinary' probability density function cause people enough confusion as it is.
I think this thread has proved that.
One of the big sources of confusion is the name "probability density" which people naturally but erroneously assume implies a density is a probability! It isn't! But you can readily get probabilities directly from its integration.
The problem is, even if you know that, like I do, if you keep calling it "probability density", your constantly telling your subconscious that it is a probability, and then, as I know from experience, you unconsciously treat it as a probability and then you wonder why the equations you make for it are spewing out complete nonsense!
And just calling it "density" is often a bad idea also because the word "density" has a wider generic meaning, asp in physics, and in some contexts you want to avoid confusing the two meanings of the word.
For these reasons, I have invented a new name for "probability density"; of just "densi". I intend to put that in my book and explain the above reasons for it, and then recommend and hope that this new word will become the new standard word to use for probability density.
Certainly since I have been using the word densi for probability density, I have stopped making the unconscious error of thinking of it as a probability and thus stopped occasionally making mathematical mistakes from that. So it works for me and I would guess it would work for most people.
Another huge potential source of total confusion:
The word 'range' can mean range of the output of the function:
http://www.purplemath.com/modules/fcns2.htm
or it can mean the range of the input of your function;
http://www.mathgoodies.com/lessons/vol8/range.html
So lets get this straight;
If a statistician speaks of the 'range' in the context of a distribution, he can either be referring to:
EITHER
The input of the function, NOT to ever be confused with its output!
OR
The output of the function, NOT to ever be confused with its input!
GREAT! Well I am glad we have that all straightened out.😕
This is why I refuse to ever use the word 'range' for the output of the function and always insist, like I will do so in my book, to use the word codomain for that output.