Math Visualization List
Calculus
First Derivative Second Derivative Integration
Determinant
2x2 & 3x3 Determinants Cramer's Rule Permutations & Inversions Permutation Definition & Expansion Properties of Determinants: Row Operations Minors & Cofactors Two-Row Laplace Expansion Vandermonde Determinant
Matrix Operations
Matrix Addition, Subtraction & Scalar Multiplication Matrix Multiplication & Power Transpose, Determinant & Adjoint Inverse Matrix & Matrix Equations Elementary Transformations, Echelon & Canonical Form
Probability & Statistics
Conditional Probability, Independence & Total Probability Discrete Probability Distributions PMF & CDF 2D Random Vectors Conditional Probability Distributions Bayes' Theorem Numerical Characteristics of Random Variables Higher Moments & Covariance Matrix Law of Iterated Expectations Independence, Mean Independence & Uncorrelated Continuous Statistical Distributions
Vector Operations
Vector Addition, Subtraction & Scalar Multiplication Linear Combination, Span & Basis Matrices as Transformations Composition of Transformations 3D Linear Transformations Determinant Inverse, Column Space & Rank Cross-Dimensional Transformations Vector Angle Cross Product Change of Basis Eigenvectors & Eigenvalues Function Vector Spaces Geometric Interpretation of Cramer's Rule Gravity Simulation
Higher Moments & Covariance Matrix

First Raw Moment ($E[X]$): The mean. It determines the horizontal position of the entire point cloud.

Second Central Moment ($Var(X)$): Measures the spread of data.

Third Central Moment ($\mu_3$): This is the highlight of this demo. When you adjust the skew slider, the point cloud becomes asymmetrical. With positive skewness, a long tail appears on the right side, and $\mu_3$ becomes positive.

Fourth Central Moment ($\mu_4$): Reflects the sharpness of the peak and thickness of the tails. Being a fourth power, it is extremely sensitive to outliers.

1st Raw Moment (Expectation)

Describes the distribution's center of gravity:

$$\nu_1 = E[X^1] = \mu$$

2nd Central Moment (Variance)

Describes the distribution's spread:

$$\mu_2 = E[(X-E[X])^2] = \sigma^2$$

3rd Central Moment (Skewness)

Describes the distribution's symmetry:

$$\mu_3 = E[(X-E[X])^3]$$

Positive = right skewed (long tail on right), Negative = left skewed.

4th Central Moment (Kurtosis)

Describes the distribution's tail thickness:

$$\mu_4 = E[(X-E[X])^4]$$

协方差矩阵 (Covariance Matrix)

For random vector $\mathbf{X} = [X, Y]^T$, the symmetric matrix is defined as:

$$\Sigma = \begin{bmatrix} Cov(X,X) & Cov(X,Y) \\ Cov(Y,X) & Cov(Y,Y) \end{bmatrix}$$

Note: Diagonals are variances, off-diagonals are covariances. This matrix is always positive semi-definite.

Dynamic Distribution Shape Observation

Adjust parameters below to observe moment changes in X-axis distribution

$\nu_1$ (Mean)

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$\mu_2$ (Variance)

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$\mu_3$ (Skewness)

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$\mu_4$ (Kurtosis)

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Real-time Covariance Matrix $\Sigma$

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