commit 3d921d5905db07ed75ab74027d913c394785688b
parent 09f92a3d4660568ec411624e9a668eae5844ddd9
Author: Andrew <andrewlaack1@gmail.com>
Date: Mon, 15 Jul 2024 22:00:30 -0500
Completed this stuff
Diffstat:
5 files changed, 70 insertions(+), 0 deletions(-)
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -36,3 +36,4 @@ The basis of linear algebra is solving systems of equations.
[[Transpose.md]]
[[NullSpace.md]]
[[Nullity.md]]
+[[Rank.md]]
diff --git a/ProbabilityMassFunction.md b/ProbabilityMassFunction.md
@@ -0,0 +1,31 @@
+:prob:
+# Probability Mass Function (PMF)
+
+L4
+
+## Notes
+
+**Definition:** A PMF describes the probability of some mapping of a [[RandomVariable.md]] from inputs to a specific output.
+
+This can be displayed as some form of bar graph.
+
+To find the PMF value for a given point we sum the probability of each input that maps to the output in question.
+
+## Example
+
+```mermaid
+
+graph LR
+a --> x
+b --> y
+c --> y
+d --> z
+```
+
+In the below example assume each connection is the function defined by the Random Variable. As such, the PMF output for x would be P(a). The PMF output for y would be P(b) + P( c ) and the output for z would be P(d).
+
+With proper notation (and assuming random variable X) we can state the above as $P_X(x) = P(A), \space P_X(y) = P(B)+P(C)$ etc.
+
+## Geometric
+
+The geometric PMF is a specific PMF where every subsequent output decreases by a given percent each time creating a form of poisson distribution.
diff --git a/RandomVariables.md b/RandomVariables.md
@@ -0,0 +1,25 @@
+:prob:
+# Random Variables
+
+L4 + Khan
+
+## Notes
+
+**Definition:** Random variables in stats and probability are functions that map processes to outcomes that depend on random events.
+
+Random variables, despite the name, are functions not variables.
+
+A random variable is any function that depends on randomness. In this way, a mapping from a value to a real number and then multiplying it by a scalar can be composed of two random variables. The first one maps to the real number takes a random input and outputs a value x times more than the input.
+
+## Formal
+
+A function from omega (sample space) to the real numbers (discrete or continuous does not matter).
+
+## Example
+
+Example:
+
+X = {1 if heads}
+ {0 if tails}
+
+Geometric random variables are random variables that result in [[ProbabilityMassFunction.md]] with a geometric shape (see PMF for more).
diff --git a/Rank.md b/Rank.md
@@ -0,0 +1,10 @@
+:lin-alg:
+# Rank
+
+Khan
+
+## Notes
+
+**Definition:** Rank, similar to [[Nullity.md]], is a way to describe the dimensionallity of the vector space generated by the columns of a matrix.
+
+[[Nullity.md]] is the same thing except specifically referring to a matrix's null space.
diff --git a/StatisticsAndProbability.md b/StatisticsAndProbability.md
@@ -59,3 +59,6 @@ L3:
- [[Independence.md]]
L4:
- [[BinomialCoefficient.md]]
+L5:
+ - [[RandomVariables.md]]
+ - [[ProbabilityMassFunction.md]]
