Vector Subspace in Linear Algebra Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Definition of a Subspace Let \( V \) be a vector space over a field \( K \) and let \( W \subseteq V \). Then \( W \) is a subspace of \( V \) if \( W \) is itself […]
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Matrix Operations with PyTorch Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Matrix Addition and Scalar Multiplication Matrix Addition: We add corresponding elements of the same-sized matrices: \( A + B = [a_{ij} + b_{ij}] \) Scalar Multiplication: Multiply each element of the matrix by the scalar value: \( kA = [k \cdot a_{ij}] \) Example:
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Binomial Distribution Data Science and A.I. Lecture Series By Bindeshwar Singh Kushwaha | PostNetwork Academy Binomial Probability Function The binomial probability function is given by: \[ P(X = k) = \binom{n}{k} p^k (1 – p)^{n – k} \] where: \( n \) = total number of trials \( k \) = number of successes
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Bernoulli Distribution Data Science and A.I. Lecture Series By Bindeshwar Singh Kushwaha | PostNetwork Academy Introduction to Bernoulli Distribution A Bernoulli trial is an experiment with only two possible outcomes: Success (1) and Failure (0). If p is the probability of success, then q = 1 – p is the probability of failure. A random
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Addition, Multiplication Theorem of Expectation and Covariance Data Science and A.I. Lecture Series By Bindeshwar Singh Kushwaha PostNetwork Academy Outline Introduction Addition Theorem of Expectation Proof of Addition Theorem Multiplication Theorem of Expectation Proof of Multiplication Theorem Covariance Introduction Expectation (or expected value) is a fundamental concept in probability and statistics. It provides a measure
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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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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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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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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
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