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These come to the same thing. Have a look at my integration diagram below.
The definite integral can be thought of as the sum of all the rectangles underneath the curve. Not the f(x)'s.
2) It is normalized; meaning that its integral over the interval of real numbers should exactly be equal to 1
The above properties tell me that each individual value of p(x) must be less than 1, if their integral over negative infinity to positive infinity is exactly 1. (correct me if I'm wrong)
Now here is the confusing part:
"... the probability density function describes the relative likelihood of a random variable (or event) having a certain value. For instance, if p(x1) = 10 and p(x2) = 100 , then the random variable with the PDF p is ten times more likely to have a value near x1 than near x2 ..."
How can we have p(x1) = 100, when the summation of all the possible p(x) values must be equal to 1? (property number 2)
I also don't get the part where it says x1 is ten times more likely than x2... shouldn't it be the other way around?
Thanks for your help