Understanding Probability: Sample Space, Sample Points, and Events Sample Space The set of all possible outcomes of a random experiment is called the Sample Space, denoted by \( S \). \( S \) is a set containing all possible outcomes of a random experiment. Examples: Tossing a single coin: \( S = \{H, T\} […]
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Deterministic to Random: The Role of Probability in AI and Data Science Introduction An experiment refers to an operation or activity that can produce some well-defined outcome(s). Types of experiments: Deterministic Experiments Random (or Probabilistic) Experiments Deterministic Experiments These experiments have a fixed outcome or result, no matter how many times they are repeated
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Spearman’s Rank Correlation Coefficient Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Need for Spearman’s Rank Correlation Coefficient In many cases, the relationship between variables is not linear, making Pearson’s correlation coefficient unsuitable. Spearman’s Rank Correlation measures the strength and direction of a monotonic relationship between two variables. It is
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Derivation of the Correlation Coefficient Data Science and A.I. Lecture Series Problem Statement Objective: Derive the formula for the correlation coefficient \( r(X, Y) \): \[ r(X, Y) = \frac{\sigma_X^2 + \sigma_Y^2 – \sigma_{X-Y}^2}{2 \sigma_X \sigma_Y}. \] Definitions: \( \sigma_X^2 \): Variance of \( X \). \( \sigma_Y^2 \): Variance of \( Y
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Prove \( -1 \leq r(X, Y) \leq 1 \) for Karl Pearson’s Correlation Coefficient Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Problem Statement Prove that: \[ -1 \leq r(X, Y) \leq 1 \] The correlation coefficient \( r(X, Y) \) is a measure of the linear relationship between
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Karl Pearson’s Correlation Coefficient Learn the step-by-step process of finding the correlation coefficient in statistics. Problem Statement Find the Karl Pearson’s coefficient of correlation between \(X\) and \(Y\) for the given data: \[ \begin{aligned} X &: 6, 2, 4, 9, 1, 3, 5, 8 \\ Y &: 13, 8, 12, 15, 9, 10, 11,
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Independence of Origin and Scale in Karl Pearson’s Correlation Coefficient Definition of Correlation Coefficient The correlation coefficient \( r(X, Y) \) is defined as: \[ r(X, Y) = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}}. \] Covariance: \[ \text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^n (x_i – \bar{X})(y_i – \bar{Y}) \] Variance of \( X \): \[ \text{Var}(X) = \frac{1}{n}
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The Definition and Calculation of The Correlation Coefficient Data Science and A.I. Lecture Series 1. Definition of Correlation Coefficient The correlation coefficient measures the strength and direction of a linear relationship between two variables. It is denoted by r, and it ranges from -1 to +1: r = +1: Perfect positive correlation. r =
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Why is Covariance Bounded? The Power of Cauchy-Schwarz Inequality Covariance and Standard Deviation Definitions: Sample Covariance: \[ \text{Cov}(X, Y) = \frac{1}{n-1} \sum_{i=1}^n (X_i – \bar{X})(Y_i – \bar{Y}) \] Sample Standard Deviations: \[ \sigma_X = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (X_i – \bar{X})^2}, \quad \sigma_Y = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (Y_i – \bar{Y})^2} \] Cauchy-Schwarz Inequality The Cauchy-Schwarz inequality states:
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Understanding Correlation: A Simplified Explanation Welcome to this post in the Data Science and A.I. Lecture Series by Bindeshwar Singh Kushwaha from PostNetwork Academy! Today, we’ll dive into correlation—a crucial concept in data science and statistics. — What is Correlation? In simple terms, correlation measures the strength and direction of the relationship between two
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