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 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 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 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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Law of Total Probability and Examples Data Science and A.I. Lecture Series By Bindeshwar Singh Kushwaha, PostNetwork Academy Partition of a Sample Space A set of events \(E_1, E_2, E_3, E_4\) represents a partition of the sample space \(S\) if: \( E_i \cap E_j = \emptyset \) for \( i \neq j \) (pairwise disjoint).
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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 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 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 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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Axiomatic Approach to Probability Data Science and A.I. Lecture Series Axiomatic Approach to Probability Definition: Let \( S \) be a sample space for a random experiment. Let \( A \) be an event that is a subset of \( S \). \( P(A) \) is called a probability function if it satisfies the
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