**Binomial Distribution and Variance Calculation**In this exercise, we will explore how to find the variance of a random variable \(X\) that follows a binomial distribution. Given that \(X\) follows \(Bin(n, p)\), the questions prompt methods to calculate the variance both algebraically and using the moment generating function.1. **Assumption:** Assume that \(X \sim Bin(n, p)\).2. **Questions:** a. **Algebraic Calculation of Variance:** Find the variance of \(X\) algebraically. A helpful hint is to first find the expected value of \(X \cdot (X - 1)\), denoted as \(E(X \cdot (X - 1))\). b. **Variance using Moment Generating Function:** Use the moment generating function \(M_X(t)\) to find the variance of \(X\), denoted as \(V(X)\).3. **Detailed Explanations:** - For part (a), to compute the variance algebraically, you will follow these steps: 1. Calculate \(E(X)\) (the mean). 2. Calculate \(E(X \cdot (X - 1))\). 3. Use the formula for variance, \(V(X) = E(X^2) - [E(X)]^2\), where \(E(X^2)\) can be derived from \(E(X \cdot (X - 1))\). - For part (b), you will need to utilize the properties of the moment generating function \(M_X(t)\) for the Binomial distribution and derive the variance from the second derivative of \(M_X(t)\) evaluated at \(t = 0\).These calculations allow one to understand the distribution and spread of the binomial random variable \(X\) around its mean, providing insight into its variability.

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Description: **Binomial Distribution and Variance Calculation**In this exercise, we will explore how to find the variance of a random variable \(X\) that follows a binomial distribution. Given that \(X\) follows \(Bin(n, p)\), the questions prompt methods to cal

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Answered: Assume that X~Bin(n, p). Find the…

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Assume that X~Bin(n, p). Find the variance of X algebraically. Hint: First find E(X*(X-1)). Use M,(1) to find V(X). a. b.

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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