9 files changed, 253 insertions, 0 deletions
diff --git a/sannolikhet/2D random variable.md b/sannolikhet/2D random variable.md
new file mode 100644
index 0000000..10c2a39
--- /dev/null
+++ b/sannolikhet/2D random variable.md
@@ -0,0 +1,56 @@
+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.
diff --git a/sannolikhet/Bayes' theorem.md b/sannolikhet/Bayes' theorem.md
new file mode 100644
index 0000000..633c26f
--- /dev/null
+++ b/sannolikhet/Bayes' theorem.md
@@ -0,0 +1,3 @@
+$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$
+
+Proof: expand the [[Conditional probability]].
diff --git a/sannolikhet/Essential formula of probability.md b/sannolikhet/Essential formula of probability.md
new file mode 100644
index 0000000..25414fc
--- /dev/null
+++ b/sannolikhet/Essential formula of probability.md
@@ -0,0 +1,9 @@
+Also known as the "happy face formula", since we are happy whenever we get to
+use it.
+
+If all outcomes are equally likely, then
+
+$$P(A) = \frac{\mathrm{N}(A)}{\mathrm{N}(S)}$$
+
+We can sometimes count using this directly, but usually we need counting
+techniques.
diff --git a/sannolikhet/Event.md b/sannolikhet/Event.md
new file mode 100644
index 0000000..1d630f0
--- /dev/null
+++ b/sannolikhet/Event.md
@@ -0,0 +1,54 @@
+An event is any subset of the [[Sample space]].
+
+Example: Given
+$$S = \{1,2,3,4,5,6\}$$
+
+then
+$$A = \{2,4\}$$
+is an event.
+
+Can be visualized with [[Venn diagram]]s.
+
+Events happen with a [[Probability]].
+
+## Special events
+
+Some special events are the empty event ($\phi$) and the [[Sample space]] itself ($S$).
+
+# Operations
+
+Just like there are normal operations (+, \*, ..) on numbers there exist
+operations on events. They work much like events on normal [[Set]]s.
+
+## Intersection
+
+$$A \cap B$$
+
+## Union
+
+$$A \cup B$$
+
+## Complement
+
+$$A'$$
+
+# Disjoint
+
+Two events are disjoint if they don't "overlap" in any way.
+
+# Independent
+
+Two events are independent if $P(A \cap B) = P(A) \cdot P(B)$.
+
+Intuition: $A$ does not affect $B$ and vice versa.
+
+$(A, B)$ independent $\Leftrightarrow$ $(A', B)$ independent.
+
+For example, given a fair dice roll and
+
+$$A = \{2\}, \ B = \{2, 3\}, \ C = S$$
+
+then $A$ and $B$ are dependent but $A$ and $C$ are independent (since the
+outcome of $A$ doesn't affect the outcome of $C$ since $P(C) = 1$.
+
+In general, $A_1, ..., A_n$ are independent if...
diff --git a/sannolikhet/Percentile.md b/sannolikhet/Percentile.md
new file mode 100644
index 0000000..a1879e9
--- /dev/null
+++ b/sannolikhet/Percentile.md
@@ -0,0 +1,5 @@
+$c$ is called the $b$-th percentile if
+
+$$P(X \le c) = b \%$$
+
+where $P(X \le x)$ denotes a [[Random variable#Continuous random variable#Probability density function]]
diff --git a/sannolikhet/Probability calculation techniques.md b/sannolikhet/Probability calculation techniques.md
new file mode 100644
index 0000000..0b0efdf
--- /dev/null
+++ b/sannolikhet/Probability calculation techniques.md
@@ -0,0 +1,16 @@
+A proper definition would probably (hah) use the [[Kolmogorov axioms]].
+
+Intuition: probability is "chance". However: something like 50% is not exact. If
+something has the probability 50% we don't expect it to happen exactly 50 times
+out of 100 tries. Rather, we expect
+
+$$\lim_{n \rightarrow \infty} \mathrm{N}(\mathrm{heads}) = 50\% \cdot \mathrm{N}(n \ \mathrm{throws})$$
+
+# Some rules
+
+$$P(S) = 1$$
+$$P(\phi) = 0$$
+$$P(A') = 1 - P(A)$$
+$$P(A_1 \cup A_2 \cup ...) = P(A_1) + P(A_2) + ...$$
+$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
+$$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C)$$
diff --git a/sannolikhet/Probability.md b/sannolikhet/Probability.md
new file mode 100644
index 0000000..90969b0
--- /dev/null
+++ b/sannolikhet/Probability.md
@@ -0,0 +1,3 @@
+In a nutshell: "What is the *probability*" that x happens?"
+
+See [[TAMS42#Probability]].
+\ No newline at end of file
diff --git a/sannolikhet/Random variable.md b/sannolikhet/Random variable.md
new file mode 100644
index 0000000..ba7e125
--- /dev/null
+++ b/sannolikhet/Random variable.md
@@ -0,0 +1,97 @@
+Definition: A random variable (or a distribution) is a numerical value
+associated with an [[Experiment]] whose value can change from one replicate of
+the experiment to another.
+
+A proper definition would need [[Probability space]] and [[Measurable function]]s.
+
+For example, given a fair dice roll,
+
+$$X = \{1,2,3,4,5,6\}$$
+
+is a random variable.
+
+$$Y = [30, 260]$$
+
+is another.
+
+Two types: Discrete random variables and continuous random variables.
+
+# [[Discrete]] random variable
+
+If the outcomes are either bounded in size or countably infinite.
+
+## Probability mass function
+
+Also known as the pmf. Every discrete random variable has a corresponding pmf.
+Denoted
+
+$$p(x) = P(X = x)$$
+
+## Table
+
+Every discrete random variable also has a corresponding table.
+
+|$X$|$x_1$|$x_2$|$...$|$x_n$|
+|--|--|--|--|--|
+|$p(x)$|$p(x_1)$|$p(x_2)$|$...$|$p(x_n)$|
+
+where
+
+$$p(x_i) \ge 0 \quad \forall i \in \{1,2,..,n\}$$
+$$\sum_{i=1}^n p(x_i) = 1$$
+
+# [[Continuous]] random variable
+
+The rest. E.g. some interval on the number line.
+
+## Probability density function
+
+Also knows as the pdf. Every continuous random variable has a corresponding pdf.
+Denoted $f(x)$ where
+
+$$\int_a^b f(x) dx = P(a \le X \le b)$$
+
+and
+
+$$f(x) \le 0 \quad \forall x$$
+$$\int_{-\infty}^\infty f(x) dx = 1$$
+
+# Cumulative distribution function
+
+Also knows as the cdf.
+
+$$F(x) = P(X \le x)$$
+
+For discrete random variables:
+
+$$F(y) = \sum_{i=1}^y p(x_i)$$
+
+And for continuous random variables:
+
+$$F(x) = \int_{-\infty}^x f(y) dy$$
+
+Here we see that
+
+$$F'(x) = f(x)$$
+
+for continuous random variables. Compare with [[Algebrans fundamentalsats]]?
+
+# Examples
+
+## Waiting time (useful model)
+
+Let $X$ be the waiting time between calls in a phone center. Assume $X$ is a
+continuous random variable with pdf
+
+$$f(x) = 2e^{-2x} \quad x \gt 0$$
+
+What is $P(X \gt 3)$?
+
+$$P(X \gt 3) = \int_3^\infty f(x) dx = \int_3^\infty 2e^{-2x}dx = e^{-6}$$
+
+In actuality,
+
+$$f(x) = 2e^{-2x} \quad x>0$$
+$$0 \ \mathrm{otherwise}$$
+
+but the 0-case is assumed.
diff --git a/sannolikhet/Sample space.md b/sannolikhet/Sample space.md
new file mode 100644
index 0000000..6e4130c
--- /dev/null
+++ b/sannolikhet/Sample space.md
@@ -0,0 +1,10 @@
+The sample space is the [[Set]] of all possible outcomes of an [[Experiment]] or
+a [[Trial]].
+
+Example: Throw a dice and observe the upper side.
+$$S = \{1,2,3,4,5,6\}$$
+
+Example: Throw two fair dice and observe their upper sides.
+$$S = \{(1,1),(1,2), ..., (1,6),(2,1), ...,(6,5),(6,6)\}$$
+
+The sample space consists of [[Event]]s.