Data Handling

Bivariate Discrete Cumulative Distribution Function

Artificial Intelligence, Data Handling, Machine Learning, Maths, Probability and Statistics, Research and Development, Statistics /

Bivariate Discrete Cumulative Distribution Function Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Joint and Marginal Distribution Functions for Discrete Random Variables Two-Dimensional Joint Distribution Function The distribution function of the two-dimensional random variable \((X, Y)\) for all real \(x\) and \(y\) is defined as: \[ F(x,y) = P(X \leq […]

Bivariate Discrete Cumulative Distribution Function Read More »

Introduction to Machine Learning

Artificial Intelligence, Data Handling, Machine Learning /

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

Introduction to Machine Learning Read More »

Central Limit Theorem (CLT) and Uniformly Minimum Variance Unbiased Estimator (UMVUE)

Artificial Intelligence, Data Handling, Research and Development, Statistics /

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: \[ Y_n = \frac{1}{n} \sum_{i=1}^{n} X_{2i-1}, \quad Z_n = \frac{1}{n} \sum_{i=1}^{n} X_{2i} \]

Central Limit Theorem (CLT) and Uniformly Minimum Variance Unbiased Estimator (UMVUE) Read More »

Random Variables and Probability Distributions

Artificial Intelligence, Data Handling, Mathematics, Maths, Probability and Statistics, Research and Development, Statistics /

Random Variables and Probability Distributions Introduction to Random Variables In many experiments, we are interested in a numerical characteristic associated with outcomes of a random experiment. A random variable (RV) is a function that assigns a numerical value to each outcome of a random experiment. Example: Consider tossing a fair die twice and defining \(

Random Variables and Probability Distributions Read More »

Bayes’ Theorem and Examples | Data Science & AI

Artificial Intelligence, Data Handling, Machine Learning, Mathematics, Maths, Probability and Statistics, Statistics /

  Bayes’ Theorem and Examples Formula The formula for Bayes’ Theorem is given by: $$ P(E_i | A) = \frac{P(E_i) P(A | E_i)}{\sum_{j=1}^{n} P(E_j) P(A | E_j)} $$ Key Terminology \(E_i\) are hypotheses or possible causes. \(P(E_i)\) is the prior probability of \(E_i\). \(P(E_i | A)\) is the posterior probability of \(E_i\). The denominator ensures

Bayes’ Theorem and Examples | Data Science & AI Read More »

Addition and Multiplicative Laws Probability Explained

Artificial Intelligence, Data Handling, Machine Learning, Mathematics, Maths, Probability and Statistics, Research and Development, Statistics /

  Problems Using Both Addition and Multiplicative Laws Data Science and A.I. Lecture Series PostNetwork Academy Probability Laws The addition law of probability states: \[ P(A \cup B) = P(A) + P(B) – P(A \cap B) \] The multiplicative law of probability for independent events states: \[ P(A \cap B) = P(A) \cdot P(B) \]

Addition and Multiplicative Laws Probability Explained Read More »

Probability

Probability of Happening at Least One Independent Event

Artificial Intelligence, Data Handling, Machine Learning, Mathematics, Maths, Probability and Statistics, Statistics /

  Probability of Happening at Least One Independent Event Data Science and A.I. Lecture Series By: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy 1. Probability of Happening at Least One Independent Event If \( A \) and \( B \) are independent events, the probability of happening at least one of the events is: \[ P(A

Probability of Happening at Least One Independent Event Read More »

Conditional Probability and Multiplicative Law, Independent Events

Artificial Intelligence, Data Handling, Mathematics, Maths, Probability and Statistics, Research and Development, Statistics /

  Conditional Probability and Multiplicative Law Data Science and A.I. Lecture Series   Conditional Probability Conditional probability represents the likelihood of an event \( A \), given that another event \( B \) has already occurred. It is defined as: \[ P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad \text{if } P(B) > 0. \] Example: Deck

Conditional Probability and Multiplicative Law, Independent Events Read More »

More on Axiomatic Approach to Probability

Artificial Intelligence, Data Handling, Machine Learning, Maths, Probability and Statistics, Research and Development, Statistics /

More on Axiomatic Approach to Probability Data Science and AI Lecture Series By Bindeshwar Singh Kushwaha Statement of the First Proof Prove: \( P(A \cap B^c) = P(A) – P(A \cap B) \) This formula expresses the probability of \( A \) occurring without \( B \). It uses the complement rule and properties of

More on Axiomatic Approach to Probability Read More »

©Postnetwork-All rights reserved.