Span and Intersection of Subspaces Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Linear Span: Definition and Properties Given vectors \( u_1, u_2, \ldots, u_m \) in a vector space \( V \), their linear span is the set of all linear combinations: \[ \text{span}(u_1, u_2, \ldots, u_m) = \{ a_1 u_1 + a_2 u_2 + […]
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Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Sir Francis Galton: Biography and Contributions Quick Facts Name: Sir Francis Galton, FRS FRAI Born: 16 February 1822, Birmingham, England Died: 17 January 1911, Haslemere, Surrey, England (Aged 88) Resting Place: Claverdon, Warwickshire, England Education: King’s College London, Trinity College Cambridge Father: Samuel Tertius Galton Relatives: Charles
What is Research? How to Write a Research Paper Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Outline What is Research? Benefits of Research Research Paper Sections Research Paper Examples Where to Publish Research Papers What is Peer Review? What is Impact Factor? Where to Start from? What is Research? Systematic investigation to discover new knowledge
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Moment Generating Function of Binomial Distribution Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Definition of Moment Generating Function The moment generating function (m.g.f.) of a random variable \( X \) is defined as: \[ M_X(t) = E(e^{tX}) \] For a continuous random variable: \[ M_X(t) = \int_{-\infty}^{\infty} e^{tx} f(x) \, dx \] For a discrete
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Solving First Order and First Degree Differential Equations Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Reducing to Homogeneous Form Consider the differential equation of the form: $$\frac{dy}{dx} = \frac{a x + b y + c}{a’ x + b’ y + c’}$$ To reduce it to homogeneous form, use the substitution: $$x = X + h,
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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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Moments of Binomial Distribution By Bindeshwar Singh Kushwaha — PostNetwork Academy Moment Definition Let \( X \sim B(n, p) \) be a binomial random variable. The \( r^\text{th} \) raw moment about origin: \( \mu_r’ = \mathbb{E}(X^r) = \sum_{x=0}^{n} x^r \cdot \mathbb{P}(X = x) \) First-order moment (mean): \( \mu_1′ = \mathbb{E}(X) \) Binomial
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Solving First Order and First Degree Differential Equations Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Example 1: Solve \( \frac{dy}{dx} = e^x \) Equation: \( \frac{dy}{dx} = e^x \) Multiply both sides by \( dx \): \( dy = e^x dx \) Integrate: \( \int dy = \int e^x dx \) Solution: \( y =
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Linear Combinations and Spanning Sets Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Introduction: In this post, we will explore the concepts of Linear Combinations and Spanning Sets in the context of vector spaces in Linear Algebra. Linear Combinations and Spanning Sets Let V be a vector space over a field K. A vector v in
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Vector and Vector Space in Linear Algebra by Bindeshwar Singh Kushwaha PostNetwork Academy Definition of ℝn The set of all n-tuples of real numbers is denoted by ℝn: u = (a1, a2, …, an) Each u is called a point or vector. The components ai are called coordinates, components, entries, or elements. The term scalar
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