Data Handling

Central Limit Theorem (CLT) and Uniformly Minimum Variance Unbiased Estimator (UMVUE)

Artificial Intelligence, Data Handling, Research and Development, Statistics /

Central Limit Theorem (CLT) and Uniformly Minimum Variance Unbiased Estimator (UMVUE) By: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Question 1 Suppose \( X_1, X_2, \dots \) is an i.i.d. sequence of random variables with common variance \( \sigma^2 > 0 \). Define: \[ Y_n = \frac{1}{n} \sum_{i=1}^{n} X_{2i-1}, \quad Z_n = \frac{1}{n} \sum_{i=1}^{n} X_{2i} \] […]

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Random Variables and Probability Distributions

Artificial Intelligence, Data Handling, Mathematics, Maths, Probability and Statistics, Research and Development, Statistics /

Random Variables and Probability Distributions Introduction to Random Variables In many experiments, we are interested in a numerical characteristic associated with outcomes of a random experiment. A random variable (RV) is a function that assigns a numerical value to each outcome of a random experiment. Example: Consider tossing a fair die twice and defining \(

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Bayes’ Theorem and Examples | Data Science & AI

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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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Addition and Multiplicative Laws Probability Explained

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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

Probability of Happening at Least One Independent Event

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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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Conditional Probability and Multiplicative Law, Independent Events

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  Conditional Probability and Multiplicative Law Data Science and A.I. Lecture Series   Conditional Probability Conditional probability represents the likelihood of an event \( A \), given that another event \( B \) has already occurred. It is defined as: \[ P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad \text{if } P(B) > 0. \] Example: Deck

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More on Axiomatic Approach to Probability

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More on Axiomatic Approach to Probability Data Science and AI Lecture Series By Bindeshwar Singh Kushwaha Statement of the First Proof Prove: \( P(A \cap B^c) = P(A) – P(A \cap B) \) This formula expresses the probability of \( A \) occurring without \( B \). It uses the complement rule and properties of

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Subjective Approach to Probability

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Subjective Approach to Probability Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha   What is the Subjective Approach? The subjective approach to probability is based on personal judgment, intuition, wisdom, and expertise. Unlike the classical or frequency-based approaches, it focuses on individual beliefs about the likelihood of an event. When to Use the

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