45 lines
1.2 KiB
Markdown
45 lines
1.2 KiB
Markdown
#Math #Probability
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# Discrete Case
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Let’s create a function expressing the probability two functions results have a sum of $s$.
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$$
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\sum _{x = -\infty}^{\infty} f(x)g(s-x)
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$$
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Let's unpack this formula. The inside of the sum finds the probability of a single case where $f$ and $g$ adds to $s$. By using a summation, we can run through every possible case that this happens.
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This operation is called a discrete convolution. Convolutions are notated as
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$$
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[f * g](s)
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$$
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# Continuous Case
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Extending the previous equation over to a continuous function, we can attain a definition like this:
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$$
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[f * g](s) = \int _{-\infty}^{\infty} f(x)g(s-x) dx
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$$
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Naturally, we'd expect this to be a probably density function of $f + g$. This is from the same effect as the discrete convolution, except we talk about this for an infinitely small point and probability densities.
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# Summary
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Convolutions return the probability or probability density of adding two functions together (this depends on the type of function you use).
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They are defined by:
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Discrete:
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$$
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[f * g](s) = \sum _{x = -\infty}^{\infty} f(x)g(s-x)
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$$
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Continuous:
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$$
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[f * g](s) = \int _{-\infty}^{\infty} f(x)g(s-x) dx
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$$ |