Mathematics

Bivariate Continuous Random Variables

Machine Learning, Mathematics, Probability and Statistics /

  Bivariate Continuous Random Variables Introduction A bivariate continuous random variable extends the concept of a single continuous random variable to two dimensions. It describes situations where two variables vary continuously and have some form of dependence or interaction. Understanding these concepts is fundamental in probability theory, statistics, and data science. Objectives Define bivariate continuous […]

Bivariate Continuous Random Variables Read More »

Continuous Cumulative Distribution Function (CDF) | Probability & Statistics

Artificial Intelligence, Machine Learning, Mathematics, Probability and Statistics, Statistics /

  Definition: Continuous CDF A continuous random variable can take an infinite number of values in a given range. The Probability Density Function (PDF) \( f(x) \) describes the likelihood of \( X \) falling within a small interval. The Cumulative Distribution Function (CDF) is given by: \[ F(x) = P[X \leq x] = \int_{-\infty}^{x}

Continuous Cumulative Distribution Function (CDF) | Probability & Statistics Read More »

Differential Equations

Calculus, Differential Equation, Mathematics /

Definition of Differential Equations A differential equation is an equation that involves one or more derivatives of an unknown function. Example: \[ \frac{dy}{dx} = 3x^2 \] Types of Differential Equations Ordinary Differential Equations (ODEs): \( \frac{dy}{dx} + 2y = x^2 \) Partial Differential Equations (PDEs): \( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0

Differential Equations Read More »

Some Questions Based on Discrete Probability Distributions

Artificial Intelligence, Machine Learning, Mathematics, Maths, Statistics /

Some Questions Based on Discrete Probability Distributions Data Science and A.I. Lecture Series   Problem 1 2 bad articles are mixed with 5 good ones. Find the probability distribution of the number of bad articles if 2 articles are drawn at random. Let \( X \) be the number of bad articles drawn. Possible values:

Some Questions Based on Discrete Probability Distributions Read More »

Discrete Random Variable and Probability Mass Function

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

  Discrete Random Variable and Probability Mass Function Data Science and A.I. Lecture Series A random variable is said to be discrete if it has either a finite or a countable number of values. Countable values are those which can be arranged in a sequence, corresponding to natural numbers. Example: Number of students present each

Discrete Random Variable and Probability Mass Function 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 »

Law of Total Probability and Examples

Artificial Intelligence, Machine Learning, Mathematics, Probability and Statistics /

Law of Total Probability and Examples Data Science and A.I. Lecture Series By Bindeshwar Singh Kushwaha, PostNetwork Academy Partition of a Sample Space A set of events \(E_1, E_2, E_3, E_4\) represents a partition of the sample space \(S\) if: \( E_i \cap E_j = \emptyset \) for \( i \neq j \) (pairwise disjoint).

Law of Total Probability and Examples 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 »

©Postnetwork-All rights reserved.