1 files changed, 97 insertions, 0 deletions
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.