Department of Statistics

Decoding Uncertainty,
Proving the Patterns.

An open academic study hub and computational workbench bridging rigorous statistical theory, pure mathematics, and scientific programming.

36 Academic Credits
03 Practical Labs
12 Core Formulas
100% Open Access
Launch Sim Lab Research Notebook
gaussian_sim.r
Interactive Normal Distribution Density
Live Simulator
0.0
1.0
Academic Ledger

Foundational Curriculum & Syllabus

A structured academic progression bridging pure mathematics, axiomatic probability, macroeconomic mechanics, and computational statistics.

Theoretical Core Axiomatic foundations across exploratory data analysis, probability measures, spectral linear algebra, and real analysis.
Computation & Labs Translating mathematical formulas into reproducible R programming, numerical simulations, and statistical data modeling.
Allied Disciplines Interdisciplinary breadth across quantitative micro/macro economics and digital information systems.
STAT

Introductory Statistics

Exploratory data analysis, stem-and-leaf plots, moments, dispersion, skewness, bivariate correlation, least-squares regression, and association of attributes.

EDA & Moments Regression Analysis Yule's Coefficient
Core Statistics
STAT

Probability & Distribution Theory

Axiomatic probability, Bayes' theorem, discrete/continuous random variables, joint distributions, MGFs, limit theorems (LLN, CLT), and parametric families.

Bayes' Theorem MGFs & Limit Theorems Standard Distributions
Core Statistics
MATH

Linear Algebra & Matrix Theory

Vector spaces, Gram-Schmidt orthogonalization, matrix operations, generalized inverses, Kronecker products, quadratic forms, and spectral eigenspaces.

Gram-Schmidt Generalized Inverses Eigenvalues
Allied Mathematics
MATH

Calculus & Real Analysis

Limits, continuity, successive derivatives, Leibnitz's rule, mean value theorems, reduction integral formulas, and Taylor series with remainders.

Leibnitz's Rule Mean Value Theorems Taylor Series
Allied Mathematics
MATH

Foundations of Mathematics

Equivalence relations, Cauchy-Chebyshev inequalities, polynomial theory of equations, 2D/3D analytic vector geometry, and linear mappings.

Classical Inequalities Descartes' Rule 3D Vector Geometry
Allied Mathematics
ECON

Principles of Economics

Microeconomic supply-demand equilibria, utility maximization, cost curves, market competition, GDP accounting, and Bangladesh agricultural economics.

Micro & Macro Public Finance Agri Economics
Allied Social Science
ICT

Information & Comm Technology

Computer architecture, operating systems, networking protocols, digital security, spreadsheets, data analytics, and modern AI landscapes.

Systems & OS Cyber Security Data & AI
All Depts Compulsory
LAB

Statistical Computing & Simulation

Translating theoretical statistics into code across 3 practical labs—R programming, Python numerical models, and statistical simulation workflows.

3 Practical Labs R & Python Monte Carlo
Computational Lab
Interactive Sim Lab

Computational Laboratory & Simulators

Real-time visual testbeds directly illustrating the mathematics from the curriculum—explore bivariate regression, probability distributions, matrix eigenspaces, and Monte Carlo asymptotics.

Bivariate OLS Linear Regression: ŷ = β₀ + β₁x Interactive Least-Squares Minimization & Pearson Correlation (r)
r = +0.80
Click anywhere on the plot above to place new data points!
Target Correlation (r): +0.80
Sample Size (N): 40
Regression Line Equation ŷ = 0.00 + 0.80x
Determination Coeff. (R²) 0.6400 (64.0%)
Squared Errors (SSE = Σ eᵢ²) 14.40
Normal Distribution N(μ, σ²) Interactive Area Under Curve P(a ≤ X ≤ b)
Continuous PDF
Mean (μ): 0.0
Std Dev (σ): 1.0
Lower Bound (a): -1.0
Upper Bound (b): 1.0
Number of Trials (n): 20
Success Probability (p): 0.50
Rate Parameter (λ): 5.0
Rate Parameter (λ): 1.0
Upper Threshold (x): 2.0
Calculated P(a ≤ X ≤ b) 0.6827 (68.3%)
Expected Value E[X] 0.00
Variance Var(X) 1.00
Linear Transformation: T(x) = Ax Grid deformation, basis vectors i' (Teal), j' (Cyan), and Invariant Axes
det(A) = 1.00
[
]
Eigenvalues (λ₁, λ₂) 2.00, 1.00
Trace tr(A) = a + d 3.00
Area Scale |det(A)| 2.00x
Dice Roll Sample Means Distribution (CLT) Convergence of sample mean x̄ₙ toward Gaussian bell curve N(μ, σ²/n)
0 Trials
Dice per Sample (n): 10
Total Sequential Steps (N): 1000
Empirical Mean (x̄) 3.50
Theoretical Mean (μ) 3.50
Sample Variance (S²) 0.29
Knowledge Logs

The Research Notebook

Intuitive derivations, simulation logs, and conceptual breakdowns designed to solidify core ideas.

LOG_001 // PROBABILITY 5 min read

The Intuition of Bayes' Theorem

Deconstructing how conditional probability updates prior beliefs in light of new evidence, resolving false-positive paradoxes.

$$P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}$$
Interactive Proof
LOG_002 // LINEAR_ALGEBRA 6 min read

Visualizing Eigenvalues Geometrically

Tracking how linear transformation matrices stretch coordinate space along invariant directional axes—the foundation of PCA.

$$A\mathbf{v} = \lambda\mathbf{v} \iff \det(A - \lambda I) = 0$$
Geometric Space
LOG_003 // LAB_COMPUTATION 4 min read

Simulating Law of Large Numbers

Running 10,000 independent Monte Carlo trials in R to observe sample means converging cleanly toward theoretical expected value.

$$\lim_{n \to \infty} P\left(|\bar{X}_n - \mu| < \varepsilon\right) = 1$$
R Code Script
LOG_004 // STAT_INFERENCE 7 min read

The Central Limit Theorem Demystified

Why the sum of independent random variables converges to a standard normal distribution regardless of underlying distribution shape.

$$\sqrt{n}\left(\bar{X}_n - \mu\right) \xrightarrow{d} \mathcal{N}(0, \sigma^2)$$
Simulation
LOG_005 // OPTIMIZATION 5 min read

Taylor Polynomial Approximations

Approximating non-linear continuous functions locally via successive higher-order derivatives and polynomial expansions.

$$f(x) \approx \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x - a)^k$$
Derivations
LOG_006 // COMPUTATION 6 min read

Monte Carlo Numerical Integration

Using pseudo-random uniform sampling to compute high-dimensional definite integrals where analytical closed forms fail.

$$\int_a^b g(x)dx \approx \frac{b-a}{N}\sum_{i=1}^N g(X_i)$$
Algorithms
Technical Stack

Core Tooling & Methods

Mathematical frameworks, computational languages, and Unix environments used throughout the curriculum.

Statistical Computing

Analytical Pipelines & Modeling
R • Python • MC
R 4.x Programming ggplot2 • tidyverse • Base R
Modeling
Python 3 NumPy • SciPy • Pandas
Scientific
Monte Carlo Simulation Bootstrap • MCMC • Stochastic
Simulation
SPSS & Stata Applied Tabulation & Inference
Analytics

Mathematical Foundations

Theoretical Rigor & Proofs
P(A|B) λv = Av ∫ f(x)
Proofs & Rigor
Probability Theory Bayes • MGFs • Asymptotics
Probability
Linear Algebra & Matrices Eigenvalues • Spectral Theory
Matrices
Calculus & Real Analysis Taylor Series • Leibnitz Rule
Analysis
Vector Geometry & Bounds 3D Dot Products • Inequalities
Geometry

Systems & Workflows

Reproducible Engineering
$ uname -r arch
Unix Pipeline
Arch Linux & Debian Rolling Kernel • Systemd Toolchains
Daily OS
Bash & Zsh Automation Shell Pipelines • POSIX Scripts
CLI / POSIX
Git & GitHub Version Control • Reproducibility
DevOps
LaTeX & KaTeX Mathematical Paper Typesetting
Publishing
Quick Reference

The Essential Formula Sheet

Searchable reference formulas with one-click LaTeX source export for study and papers.

Showing 12 of 12 Core Formulas
STAT

Pearson Correlation & OLS Slope

$$r = \frac{S_{xy}}{S_x S_y}, \quad \hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$$

Quantifies linear association strength and determines the least-squares regression slope minimizing squared residuals.

STAT

Normal (Gaussian) Distribution PDF

$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

Continuous probability distribution symmetrical around mean \(\mu\) with dispersion governed by standard deviation \(\sigma\).

STAT

Binomial & Poisson PMFs

$$P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}, \quad P(Y=k) = \frac{\lambda^k e^{-\lambda}}{k!}$$

Discrete probability mass functions for \(n\) independent Bernoulli trials and rare event arrival counts at rate \(\lambda\).

STAT

Bayes' Rule (Total Probability)

$$P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B \mid A)P(A) + P(B \mid A^c)P(A^c)}$$

Calculates updated posterior belief of event \(A\) upon observing evidence \(B\) using exhaustive partition likelihoods.

STAT

Expectation & Variance Scaling

$$\mathbb{E}[aX + b] = a\mathbb{E}[X] + b, \quad \mathrm{Var}(aX + b) = a^2\mathrm{Var}(X)$$

Fundamental algebraic properties governing linear shifts and scalar amplifications on random variable moments.

MATH

Characteristic Equation (Eigenvalues)

$$\det(A - \lambda I) = 0 \iff A\mathbf{v} = \lambda\mathbf{v}$$

Determines scalar eigenvalues \(\lambda\) and invariant directional eigenvectors \(\mathbf{v}\) for square linear transformation \(A\).

STAT / LAB

Central Limit Theorem (Standardized)

$$Z_n = \frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \mathcal{N}(0, 1)$$

As sample size \(n \to \infty\), the normalized mean of i.i.d. random variables converges in distribution to standard normal.

MATH

Taylor Series Expansion

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n$$

Represents smooth differentiable functions as infinite polynomial series centered at expansion anchor point \(a\).

MATH

Leibnitz's Product Derivative Rule

$$(uv)^{(n)} = \sum_{k=0}^n \binom{n}{k} u^{(n-k)} v^{(k)}$$

Computes the \(n\)-th successive derivative of the product of two functions \(u(x)\) and \(v(x)\) via binomial expansion.

MATH

Cauchy-Schwarz & Chebyshev

$$(\mathbf{u}\cdot\mathbf{v})^2 \le \|\mathbf{u}\|^2 \|\mathbf{v}\|^2, \quad P(|X-\mu| \ge k\sigma) \le \frac{1}{k^2}$$

Geometric vector inner-product bound and probabilistic dispersion ceiling for arbitrary random variables.

STAT

Yule's Coefficient of Association

$$Q = \frac{ad - bc}{ad + bc}, \quad -1 \le Q \le +1$$

Measures the degree of association between two dichotomous attributes in a \(2 \times 2\) contingency table.

LAB

Monte Carlo Numerical Integration

$$\int_a^b g(x)dx \approx \frac{b-a}{N}\sum_{i=1}^N g(U_i), \quad U_i \sim \mathcal{U}(a, b)$$

Approximates definite integrals over \([a, b]\) by evaluating sample mean over \(N\) uniformly generated pseudo-random points.

Knowledge Hub & Connect

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Open Academic Dialogue

Whether exploring statistical distributions, linear algebra derivations, or collaborating on computational simulation scripts, inquiries and study discussions are always welcome.

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