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Anonymous at Mon, 25 Mar 2024 18:36:14 UTC No. 16096459
stupid question about integrals
What is the difference between an integral defined for [math]dP(x)[/math] and [math]P(x)dx [/math]?
I see both forms in probability contexts, my intuition is that the dP(x) is a more general definition based on measure theory but in "common" context is that the same as P(x)dx ?
Anonymous at Mon, 25 Mar 2024 18:38:34 UTC No. 16096465
>>16096459
It is the same as P(x)dx in the case where the density function P(x) exists and is properly defined.
The reason people like to use probability measures is that there are many cases where a distribution (i.e., an integral of some function w.r.t the probability measure dP) is defined, where a density may not be.
Btw, if you want a fantastic series about probability theory that is on YouTube, check out Todd Kemp's Probability Theory lectures on YouTube. They are great.
Anonymous at Mon, 25 Mar 2024 19:41:01 UTC No. 16096570
>>16096459
>my intuition is that the dP(x) is a more general definition based on measure theory but in "common" context is that the same as P(x)dx ?
correct for most practical situations
for me, the "common" context is one where i'm actually computing a numerical solution with well-defined functions
i'll use the general definition for making property based/categorical statements
Anonymous at Mon, 25 Mar 2024 20:11:26 UTC No. 16096606
>>16096459
[math]dP(x) = P'(x)dx[/math], not [math]P(x)dx[/math]
The difference is that the former is defined even when P is not differentiable. Look up the "Riemann-Stieltjes integral" for more details.
Anonymous at Mon, 25 Mar 2024 20:57:30 UTC No. 16096661
One is Riemann integral, next is Lebesgue integral. These are often interchangeable but sometimes not.
Anonymous at Mon, 25 Mar 2024 23:13:51 UTC No. 16096822
>>16096459
>dP(x)
is a riemann-stieltjes integral.
it's better behaved than a riemann integral, which is important for the distributions encountered in probability