commit 1a7f4b48ba1c4b0c8f11aba35b0557a2a97e6334
parent c9f51d214913638ec2c86f40cd7ede6cc6aabc87
Author: Andrew <andrewlaack1@gmail.com>
Date: Wed, 17 Jul 2024 17:54:13 -0500
Took some notes
Diffstat:
18 files changed, 114 insertions(+), 42 deletions(-)
diff --git a/Codomain.md b/Codomain.md
@@ -0,0 +1,16 @@
+:math310: :lin-alg:
+# Codomain
+
+Khan
+
+## Notes
+
+**Definition:** The codomain of a function is a set that contains all possible mappings from the domain of inputs to outputs. This set can also contain values that are not mapped to from the domain by the function.
+
+See [[Range.md]] for only the subset of the codomain that is mapped to.
+
+Defined formally, we can have any codomain C(f) that fulfills the following where D is the domain of the function f:
+
+$C(f) \supseteq \{y \space | \space \exists x \in D \text{ such that } f(x) = y\}$
+
+Despite the openness of this, we often use a predefined set as the set considered the codomain, but it can be any set we choose to define that contains the range of the function.
diff --git a/ColumnSpace.md b/ColumnSpace.md
@@ -1,7 +1,7 @@
:lin-alg:
# Column Space
-Lec 1
+Khan
## Notes
diff --git a/DistanceCalculation.md b/DistanceCalculation.md
@@ -1,7 +1,7 @@
:lin-alg:
# Distance Calculation
-Reading
+Khan
## Notes
diff --git a/DotProduct.md b/DotProduct.md
@@ -1,7 +1,7 @@
:cs331: :lin-alg:
# Dot Product
-CS331 + Linear Algebra
+CS331 + Khan
## Notes
diff --git a/Expectation.md b/Expectation.md
@@ -0,0 +1,18 @@
+:prob:
+# Expectation (Expected Value of Random Variable)
+
+L6
+
+## Notes
+
+**Definition:** The expected value of a PMF is the weighted average of output.
+
+This is calculated by summing the probabilities of each output multiplied by the output value. This will be the 'middle' of the sample space (weighted average).
+
+This is denoted by the function E[] where the inside is the random variable that is being predicted upon.
+
+Interesting note, the signed difference from the expectation will always be zero hence why we square them to find the variance and subsequently the standard deviation.
+
+## Conditional (L6)
+
+Conditional expectations are just expectations, but they are in reference to the conditional PMF instead of the original one.
diff --git a/Invertible.md b/Invertible.md
@@ -1,7 +1,7 @@
:lin-alg:
# Invertible (Matrix)
-Reading
+Khan
## Notes
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -7,33 +7,39 @@ The basis of linear algebra is solving systems of equations.
## Links
-[[Matrix.md]]
-[[LinearEquations.md]]
-[[SystemsOfEquations.md]]
-[[LinearCombination.md]]
-[[ColumnSpace.md]]
-[[DistanceCalculation.md]]
-[[DotProduct.md]]
-[[VectorMatrixMultipication.md]]
-[[Invertible.md]]
-[[UnitVector.md]]
-[[Span.md]]
-[[LinearIndependence.md]]
-[[LinearSubspace.md]]
-[[Closure.md]]
-[[BasisOfSubspace.md]]
-[[AngleBetweenVectors.md]]
-[[LawOfCosines.md]]
-[[EquationOfAPlane.md]]
-[[CrossProduct.md]]
-[[Arcsin.md]]
-[[Arccos.md]]
-[[TripleProductExpansion.md]]
-[[NormalVector.md]]
-[[DistanceToPlane.md]]
-[[PlaneToPlaneDistance.md]]
-[[ReducedRowEchelonForm.md]]
-[[Transpose.md]]
-[[NullSpace.md]]
-[[Nullity.md]]
-[[Rank.md]]
+
+Khan Unit 1 (mostly):
+ - [[Matrix.md]]
+ - [[LinearEquations.md]]
+ - [[SystemsOfEquations.md]]
+ - [[LinearCombination.md]]
+ - [[ColumnSpace.md]]
+ - [[DistanceCalculation.md]]
+ - [[DotProduct.md]]
+ - [[VectorMatrixMultipication.md]]
+ - [[Invertible.md]]
+ - [[UnitVector.md]]
+ - [[Span.md]]
+ - [[LinearIndependence.md]]
+ - [[LinearSubspace.md]]
+ - [[Closure.md]]
+ - [[BasisOfSubspace.md]]
+ - [[AngleBetweenVectors.md]]
+ - [[LawOfCosines.md]]
+ - [[EquationOfAPlane.md]]
+ - [[CrossProduct.md]]
+ - [[Arcsin.md]]
+ - [[Arccos.md]]
+ - [[TripleProductExpansion.md]]
+ - [[NormalVector.md]]
+ - [[DistanceToPlane.md]]
+ - [[PlaneToPlaneDistance.md]]
+ - [[ReducedRowEchelonForm.md]]
+ - [[Transpose.md]]
+ - [[NullSpace.md]]
+ - [[Nullity.md]]
+ - [[Rank.md]]
+
+Khan Unit 2:
+ - [[Codomain.md]]
+ - [[Range.md]]
diff --git a/LinearCombination.md b/LinearCombination.md
@@ -1,7 +1,7 @@
:lin-alg:
# Linear Combination
-Lec 1
+Khan
## Notes
diff --git a/LinearEquations.md b/LinearEquations.md
@@ -1,7 +1,7 @@
:lin-alg:
# Linear Equations
-Lec 1
+Khan
## Notes
diff --git a/Matrix.md b/Matrix.md
@@ -1,7 +1,7 @@
:lin-alg:
# Matrix
-L1
+Khan
## Notes
diff --git a/ProbabilityMassFunction.md b/ProbabilityMassFunction.md
@@ -35,3 +35,9 @@ This is denoted by the function E[] where the inside is the random variable that
## 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.
+
+The geometric PMF is **memoryless** in that regardless of the step you start on, the future probabilities will be the same because all conditionals result in no added information thus they are independent probabilities.
+
+## Conditional (L6)
+
+Conditional PMFs are just PMFs but they have a specified even that occurred. In these instances we simply resize the sample space accordingly and then recalculate probabilities.
diff --git a/Range.md b/Range.md
@@ -0,0 +1,12 @@
+:lin-alg: :math310:
+# Range
+
+Khan
+
+## Notes
+
+**Definition:** The range of a function is the set of all possible outputs of the function given the domain of the function.
+
+Formally we can state it as the following where D is the domain of the function and R is the range of the input function:
+
+$R(f) = \{y \space | \space \exists x \in D \text{ such that } f(x)=y\}$
diff --git a/StandardDeviation.md b/StandardDeviation.md
@@ -7,5 +7,8 @@ Stats D2
**Definition:** This is the average difference between each value in a dataset and the mean of the dataset.
+See also [[Variance.md]] which is the squared value. As such, to find the standard deviation of some random variable X we can do the following:
-See also [[Variance.md]] which is the squared value.
+std.dev = sqrt(var(X))
+
+Where var$(X) = \sum_x(x-E[X])^2p_X(x)$
diff --git a/StatisticsAndProbability.md b/StatisticsAndProbability.md
@@ -62,3 +62,8 @@ L4:
L5:
- [[RandomVariables.md]]
- [[ProbabilityMassFunction.md]]
+L6:
+ - [[ProbabilityMassFunction.md]]
+ - [[Expectation.md]]
+ - [[Variance.md]]
+ - [[StandardDeviation.md]]
diff --git a/SystemsOfEquations.md b/SystemsOfEquations.md
@@ -1,7 +1,7 @@
:lin-alg:
# Systems of Equations
-Lec 1
+Khan
## Notes
diff --git a/UnitVector.md b/UnitVector.md
@@ -1,7 +1,7 @@
:lin-alg:
# Unit Vector
-Reading
+Khan
## Notes
diff --git a/Variance.md b/Variance.md
@@ -9,12 +9,18 @@ Stats D2
This can be shown as follows for X:
-Var(X) = |X|^-1 * sum((x - mean)^2)
+var$(X) = \sum_x(x-E[X])^2p_X(x)$
+
+For this it is paramount to understand that the multiplication by the weight goes outside of the squared area.
Shown above, find the difference between each value and the mean, square it to get a positive, and then sum the values. We then average it by multiplying by 1 over the cardinality of X.
If we take the square root of the variance we then have the [[StandardDeviation.md]]
+Additionally, the std deviation, given our definition of variance, is equal to sqrt(var(X)) given that the variance of the random variable X is squared.
+
+Important: When referring to values with units the variance will be units^2 hence standard deviation is often better in this regard because it is simply units.
+
## Notes (ML)
**Definition:** Variance is error cause by an oversensitive model (sensitive to variance/outliers).
diff --git a/VectorMatrixMultipication.md b/VectorMatrixMultipication.md
@@ -1,7 +1,7 @@
:lin-alg:
# Vector Matrix Multiplication
-Reading
+Khan
## Notes
