Research and Development

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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Continuous Random Variable and Probability Density Function

Artificial Intelligence, Machine Learning, Probability and Statistics, Research and Development, Statistics /

  Continuous Random Variable and Probability Density Function Data Science and A.I. Lecture Series Continuous Random Variable and Probability Density Function A random variable is continuous if it can take any real value within a given range. Instead of probability mass function, we use probability density function (PDF), denoted by \( f(x) \). The probability

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Discrete Random Variable and Probability Mass Function

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

  Discrete Random Variable and Probability Mass Function Data Science and A.I. Lecture Series A random variable is said to be discrete if it has either a finite or a countable number of values. Countable values are those which can be arranged in a sequence, corresponding to natural numbers. Example: Number of students present each

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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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Addition and Multiplicative Laws Probability Explained

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

  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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Conditional Probability and Multiplicative Law, Independent Events

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

  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

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

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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Venn Diagrams

Algebra, Artificial Intelligence, Research and Development, Statistics /

Venn Diagrams – Data Science and AI Lecture Series Welcome to our Data Science and AI Lecture Series! In this post, we’ll dive into the world of Venn Diagrams, an essential tool in set theory that simplifies understanding the relationships between sets. Whether you’re studying mathematics, data science, or AI, mastering concepts like intersections, unions,

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Subjective Approach to Probability

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

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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