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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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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Binomial Distribution Overview A discrete random variable \( X \) is said to follow a binomial distribution with parameters \( n \) and \( p \) if it takes on values \( x = 0, 1, 2, …, n \) with the probability mass function: \[ P(X = x) = \binom{n}{x} p^x q^{n – x},
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Binomial Distribution Data Science and A.I. Lecture Series By Bindeshwar Singh Kushwaha | PostNetwork Academy Binomial Probability Function The binomial probability function is given by: \[ P(X = k) = \binom{n}{k} p^k (1 – p)^{n – k} \] where: \( n \) = total number of trials \( k \) = number of successes
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Addition, Multiplication Theorem of Expectation and Covariance Data Science and A.I. Lecture Series By Bindeshwar Singh Kushwaha PostNetwork Academy Outline Introduction Addition Theorem of Expectation Proof of Addition Theorem Multiplication Theorem of Expectation Proof of Multiplication Theorem Covariance Introduction Expectation (or expected value) is a fundamental concept in probability and statistics. It provides a measure
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Mathematical Expectation Data Science and A.I. Lecture Series By Bindeshwar Singh Kushwaha – PostNetwork Academy Introduction This unit explores the expectation of a random variable. Expectation provides a measure of central tendency in probability distributions. Expectation is useful in both discrete and continuous probability distributions. Problems and examples help in understanding practical applications. Objectives Define
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Bivariate Continuous Random Variables Introduction A bivariate continuous random variable extends the concept of a single continuous random variable to two dimensions. It describes situations where two variables vary continuously and have some form of dependence or interaction. Understanding these concepts is fundamental in probability theory, statistics, and data science. Objectives Define bivariate continuous
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Bivariate Discrete Cumulative Distribution Function Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Joint and Marginal Distribution Functions for Discrete Random Variables Two-Dimensional Joint Distribution Function The distribution function of the two-dimensional random variable \((X, Y)\) for all real \(x\) and \(y\) is defined as: \[ F(x,y) = P(X \leq
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Bivariate Discrete Random Variables Data Science and A.I. Lecture Series By Bindeshwar Singh Kushwaha, PostNetwork Academy Definition Let \( X \) and \( Y \) be two discrete random variables defined on the sample space \( S \) of a random experiment. Then, the function \( (X, Y) \) defined on the same sample space
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Definition: Continuous CDF A continuous random variable can take an infinite number of values in a given range. The Probability Density Function (PDF) \( f(x) \) describes the likelihood of \( X \) falling within a small interval. The Cumulative Distribution Function (CDF) is given by: \[ F(x) = P[X \leq x] = \int_{-\infty}^{x}
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