- #1
Kreizhn
- 743
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Hello everyone,
I have a quick theoretical question regarding probability. If you answer, I would appreciate it if you would be as precise as possible about terminology.
Here is the problem: I'm working on some physics problems that do probability in abstract spaces and the author freely moves between calling some poorly defined function f a density, a measure, and a distribution. From my knowledge of measure theory, these are all very different things, though are all inter-related.
In particular, the author talks about having two objects, say [itex] x_1 [/itex] sampled from [itex] f_1 [/itex] and [itex] x_2 [/itex] sampled from [itex] f_2 [/itex]. He wants to calculate the joint sampling distribution (?) of [itex] x = x_1 \cdot x_2 [/itex] which is defined via convolution
[tex] (f_1 \star f_2)(x) = \int f_1(x \cdot x_2^{-1}) f_2(x_2) d x_2 [/tex]
Where I hope it's clear that we're using multiplicative notation rather than the classical additive notation.
What it comes down to is that I'm trying to figure out what the author really means when talking about f. Is f a density or a distribution?
http://en.wikipedia.org/wiki/Convolution#Applications" says that densities obey convolution.
It would seem to me that this must be a density since if it were a distribution we would need to integrate over "sections" or at least non-zero measure sets.
Anyway, I really just want an answer as to whether densities, distributions, or both obey convolution. Thanks.
I have a quick theoretical question regarding probability. If you answer, I would appreciate it if you would be as precise as possible about terminology.
Here is the problem: I'm working on some physics problems that do probability in abstract spaces and the author freely moves between calling some poorly defined function f a density, a measure, and a distribution. From my knowledge of measure theory, these are all very different things, though are all inter-related.
In particular, the author talks about having two objects, say [itex] x_1 [/itex] sampled from [itex] f_1 [/itex] and [itex] x_2 [/itex] sampled from [itex] f_2 [/itex]. He wants to calculate the joint sampling distribution (?) of [itex] x = x_1 \cdot x_2 [/itex] which is defined via convolution
[tex] (f_1 \star f_2)(x) = \int f_1(x \cdot x_2^{-1}) f_2(x_2) d x_2 [/tex]
Where I hope it's clear that we're using multiplicative notation rather than the classical additive notation.
What it comes down to is that I'm trying to figure out what the author really means when talking about f. Is f a density or a distribution?
http://en.wikipedia.org/wiki/Convolution#Applications" says that densities obey convolution.
It would seem to me that this must be a density since if it were a distribution we would need to integrate over "sections" or at least non-zero measure sets.
Anyway, I really just want an answer as to whether densities, distributions, or both obey convolution. Thanks.
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