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Let X1, X2, ..., Xn be independent random variables, Xi ∼ Binomial(ni, p), i = 1, ..., n. Find the probability distribution of the sum Pni=1 Xi

Let X1, X2, ..., Xn be independent random variables, Xi ∼ Binomial(ni, p), i = 1, ..., n. Find the probability distribution of the sum Pni=1 Xi

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Let X1, X2, ..., Xn be independent random variables, Xi ∼ Binomial(ni, p), i = 1, ..., n. Find the probability distribution of the sum Pni=1 Xi

 

**Title: Understanding the Probability Distribution of Sums of Binomial Random Variables** **Introduction:** In this section, we explore the concept of determining the probability distribution for the sum of independent random variables, each following a Binomial distribution. This is a fundamental area of probability theory with significant implications in statistics and various applied fields. **Problem Statement:** Let \( X_1, X_2, \ldots, X_n \) be independent random variables such that each \( X_i \) follows a Binomial distribution with parameters \( n_1 \) and \( p \). This is denoted as \( X_i \sim \text{Binomial}(n_1, p) \) for \( i = 1, \ldots, n \). Our task is to find the probability distribution of their sum \( \sum_{i=1}^n X_i \). **Conceptual Overview:** The sum of independent binomial random variables, where each has the same probability \( p \) but potentially different number of trials, results in another binomial distribution. Specifically, the sum \( \sum_{i=1}^n X_i \) follows a binomial distribution with parameters equaling the sum of the individual trials and a common probability \( p \). **Implications:** Understanding this concept aids in statistical calculations and can be applied in scenarios ranging from quality testing to biological experiments where cumulative probabilities are assessed. **Concluding Thoughts:** The ability to determine the distribution of sums of binomially distributed variables is a powerful tool in both theoretical investigations and practical applications.
Transcribed Image Text:**Title: Understanding the Probability Distribution of Sums of Binomial Random Variables** **Introduction:** In this section, we explore the concept of determining the probability distribution for the sum of independent random variables, each following a Binomial distribution. This is a fundamental area of probability theory with significant implications in statistics and various applied fields. **Problem Statement:** Let \( X_1, X_2, \ldots, X_n \) be independent random variables such that each \( X_i \) follows a Binomial distribution with parameters \( n_1 \) and \( p \). This is denoted as \( X_i \sim \text{Binomial}(n_1, p) \) for \( i = 1, \ldots, n \). Our task is to find the probability distribution of their sum \( \sum_{i=1}^n X_i \). **Conceptual Overview:** The sum of independent binomial random variables, where each has the same probability \( p \) but potentially different number of trials, results in another binomial distribution. Specifically, the sum \( \sum_{i=1}^n X_i \) follows a binomial distribution with parameters equaling the sum of the individual trials and a common probability \( p \). **Implications:** Understanding this concept aids in statistical calculations and can be applied in scenarios ranging from quality testing to biological experiments where cumulative probabilities are assessed. **Concluding Thoughts:** The ability to determine the distribution of sums of binomially distributed variables is a powerful tool in both theoretical investigations and practical applications.
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