Vectors in \(\mathbb{R}^n\) and \(\mathbb{C}^n\) Introduction to Vectors A vector is a mathematical object that has both magnitude and direction. Vectors are essential in physics, engineering, and mathematics. They can be represented in different dimensions, such as real number space \(\mathbb{R}^n\) and complex number space \(\mathbb{C}^n\). Visualization of Vectors in \(\mathbb{R}^3\) Consider a three-dimensional space […]
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The Rise of Generative AI: Overview Unlike traditional AI systems that rely on predefined rules, generative AI models use vast datasets and deep learning techniques to generate novel and contextually relevant outputs. This transformative capability is reshaping industries such as content creation, education, healthcare, and entertainment. How Generative AI Works At its core, generative
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Supervised Learning in Machine Learning Introduction Supervised learning is a machine learning technique where a model learns from labeled data. The dataset consists of input features \((x_1, x_2, x_3)\) and an output label \(y\). The goal is to find a function that maps inputs to correct outputs. Features and Labels in a Dataset A
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Introduction to Python Programming Fundamentals and Examples Welcome to this introduction to Python programming. This post covers the fundamentals of Python, including its definition, programming concepts, and example codes. What is a Program? A program is a sequence of instructions that a computer can execute. Programs help automate tasks and solve problems. They consist of
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Introduction to Machine Learning Definition and Types Welcome to this detailed introduction to Machine Learning. This post explores the fundamental definitions, types of machine learning, and their mathematical representations. What is Machine Learning? What is Machine Learning? What are the different types of Machine Learning? How can we mathematically define each type? Definition of Machine
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Some Questions Based on Continuous Probability Distributions Question Compute the conditional probability: \[ P\left(X > \frac{3}{4} \mid X > \frac{1}{2}\right) \] Theory Behind Solution The conditional probability formula: \[ P(A | B) = \frac{P(A \cap B)}{P(B)} \] For continuous random variables, probability is computed using integration. Understanding Probability Density Functions A probability density function (p.d.f.)
A Beginner’s Guide to the Poisson Distribution with Real-World Examples Probability theory plays a crucial role in various real-world scenarios, particularly when dealing with random events that occur over a specific interval of time or space. One of the most widely used probability distributions for such scenarios is the Poisson distribution. This distribution helps
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Some Questions of Linear Algebra: Linear Transformation Definition: Linear Transformation A linear transformation \(T: \mathbb{R}^n \to \mathbb{R}^m\) is a function that satisfies: Additivity: \(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\) for all \(\mathbf{u}, \mathbf{v} \in \mathbb{R}^n\). Homogeneity: \(T(c \mathbf{v}) = c T(\mathbf{v})\) for all \(\mathbf{v} \in \mathbb{R}^n\) and scalars \(c\). Numerical Example: Linear Transformation
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Central Limit Theorem (CLT) and Uniformly Minimum Variance Unbiased Estimator (UMVUE) By: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Question 1 Suppose \( X_1, X_2, \dots \) is an i.i.d. sequence of random variables with common variance \( \sigma^2 > 0 \). Define: \[ Y_n = \frac{1}{n} \sum_{i=1}^{n} X_{2i-1}, \quad Z_n = \frac{1}{n} \sum_{i=1}^{n} X_{2i} \]
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Continuous Random Variable and Probability Density Function Data Science and A.I. Lecture Series Continuous Random Variable and Probability Density Function A random variable is continuous if it can take any real value within a given range. Instead of probability mass function, we use probability density function (PDF), denoted by \( f(x) \). The probability
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