Moments and Other Measures in Terms of Expectations Data Science and A.I. Lecture Series By Bindeshwar Singh Kushwaha – PostNetwork Academy Moments The \( r^{th} \) order moment about any point \( A \) of a variable \( X \) is given by: For discrete variables: \[ \mu_r’ = \sum_{i=1}^{n} p_i (x_i – A)^r […]
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Mathematical Expectation Data Science and A.I. Lecture Series By Bindeshwar Singh Kushwaha – PostNetwork Academy Introduction This unit explores the expectation of a random variable. Expectation provides a measure of central tendency in probability distributions. Expectation is useful in both discrete and continuous probability distributions. Problems and examples help in understanding practical applications. Objectives Define
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Operators in Python Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha | Institute: PostNetwork Academy Introduction Operators in Python are special symbols that perform computations on operands. Python provides various types of operators: Arithmetic Operators Relational Operators Logical Operators Bitwise Operators Assignment Operators Membership Operators Identity Operators Arithmetic Operators a = 10
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Python Input and Output Understanding Input and Output in Python Author: Bindeshwar Singh Kushwaha | Institute: PostNetwork Academy What is Input and Output? In Python, input and output refer to the mechanisms by which a program interacts with users. Input: Data provided by the user using the input() function. Output: Information displayed using the
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Variables and Assignment in Python What is a Variable? A variable is a name that refers to a value stored in memory. It acts as a reference, not a storage container. Unlike other languages, Python variables do not store values but rather point to objects in memory. Example: x = 10 # x refers
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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
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IBM Research and Hugging Face Introduce SmolDocling: A Compact Vision-Language Model for Document Conversion IBM Research and Hugging Face have unveiled SmolDocling, an ultra-compact vision-language model designed for end-to-end document conversion. Unlike traditional models that rely on large foundational architectures or complex pipelines, SmolDocling offers a lightweight, efficient solution for processing entire documents while preserving
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
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Bivariate Discrete Random Variables Data Science and A.I. Lecture Series By Bindeshwar Singh Kushwaha, PostNetwork Academy Definition Let \( X \) and \( Y \) be two discrete random variables defined on the sample space \( S \) of a random experiment. Then, the function \( (X, Y) \) defined on the same sample space
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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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