**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.

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**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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