From c4950dfb23d1fe8cca7cb523d9ae4ea4b4dda2d5 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Gustav=20S=C3=B6rn=C3=A4s?= <gustav@sornas.net>
Date: Fri, 21 Jan 2022 13:28:45 +0100
Subject: move files

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+A [[Random variable]] with 2 "parts".
+
+Works the same for variables with even higher dimension.
+
+# [[Discrete]] 2D random variable
+
+## Joint probability mass function (pmf)
+
+$$p(x,y) = P(X=x \cap Y=y) \quad \left( = P(X=x, Y=y) \right)$$
+
+## Joint table
+
+Much like the normal joint table but an actual table instead of a single line.
+
+We can get the respective 1D tables by adding rows or columns (depending on
+which variable) together.
+
+$$p_X(x) \ \mathrm{of} \ X := p_X(x) = p(x,y_1) + ... + p(x, y_n)$$
+
+## Marginal pdf
+
+# [[Continuous]] 2D random variable
+
+## Joint probability distribution function (pdf)
+
+1D is integrated over the number line, so 2D is integrated over $D$.
+
+$$? = \iint_D f(x,y) dxdy = P((X, Y) \in D)$$
+
+where $D$ is any [[Borel set]] on $\mathbb{R}^2$.
+
+We don't usually draw this graph since it is in 3D (unpleasant).
+
+Instead, we draw the non-trivial domain (all non-zero values).
+
+$$p(x,y) \ge 0, \ \int_{-\infty}^\infty \left( \int_{-\infty}^\infty f(x,y) dx \right) dy = 1$$
+
+## Marginal pdf
+
+$$f_X(x) \ \mathrm{of} \ X := \int_{-\infty}^\infty f(x,y)dy$$
+
+Similar for $f_Y(y)$.
+
+Bounds might be weird.
+
+# Independance
+
+$X$ and $Y$ are independent if
+
+$$p(x,y) = p_X(x) \cdot p_Y(y) \qquad \mathrm{(discrete)}$$
+$$f(x,y) = f_X(x) \cdot f_Y(y) \qquad \mathrm{(continuous)}$$
+
+Much the same as independence for [[Event]]s.
+
+Check by multiplying marginal pmf/pdf or if something "looks" dependant. Check
+the intuition.
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