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} […]
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Ordinary Least Squares (OLS) Regression Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Dataset of a Company X (Budget) Y (Sales) 1 2 2 2.8 3 3.6 4 4.5 5 5.1 Description: The dataset represents the relationship between advertising budget (\(X\)) and sales revenue (\(Y\)). The company wants to analyze how the budget affects sales using
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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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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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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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:
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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
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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).
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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: \[ P(A \cap B) = P(A) \cdot P(B) \]
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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
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