Initial commit

Convolutions.md

@@ -0,0 +1,45 @@
+#Math #Probability
+
+# Discrete Case
+
+Let’s create a function expressing the probability two functions results have a sum of $s$.
+
+$$
+\sum _{x = -\infty}^{\infty} f(x)g(s-x)
+$$
+
+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.
+
+This operation is called a discrete convolution. Convolutions are notated as
+
+$$
+[f * g](s)
+$$
+
+# Continuous Case
+
+Extending the previous equation over to a continuous function, we can attain a definition like this:
+
+$$
+[f * g](s) = \int _{-\infty}^{\infty} f(x)g(s-x) dx
+$$
+
+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.
+
+# Summary
+
+Convolutions return the probability or probability density of adding two functions together (this depends on the type of function you use).
+
+They are defined by:
+
+Discrete:
+
+$$
+[f * g](s) = \sum _{x = -\infty}^{\infty} f(x)g(s-x)
+$$
+
+Continuous:
+
+$$
+[f * g](s) = \int _{-\infty}^{\infty} f(x)g(s-x) dx
+$$