Find the mean and variance for the probability distribution given by f(x)= 2x k(k+1)* -,x=1,2,3,...k
Find the mean and variance for the probability distribution given by f(x)= 2x k(k+1)* -,x=1,2,3,...k
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**Title: Calculating the Mean and Variance of a Probability Distribution**
**Introduction:**
In this section, we will determine the mean and variance for a given probability distribution.
**Given Probability Distribution:**
The probability distribution is defined by the function:
\[ f(x) = \frac{2x}{k(k+1)}, \quad x = 1, 2, 3, \ldots, k \]
**Explanation:**
- \( f(x) \) represents the probability function for values \( x \) ranging from 1 to \( k \).
- The function is a probability distribution, where the probabilities are dependent on \( x \) and the constant \( k \).
**Objective:**
Our objective is to find the mean (expected value) and variance of this probability distribution.
**Steps to Understand:**
1. **Mean (Expected Value):**
- The mean is calculated using the formula:
\[ E(X) = \sum_{x=1}^{k} x \cdot f(x) \]
2. **Variance:**
- Variance is calculated as:
\[ Var(X) = E(X^2) - (E(X))^2 \]
- Where \( E(X^2) = \sum_{x=1}^{k} x^2 \cdot f(x) \)
In the next sections, we will derive these values using the given probability distribution function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2844e9b7-fdae-4ffb-b3b7-8f54bdb6d500%2F5abd5245-3f97-42bf-8fef-43a86dcaf138%2Fusmq55w_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Calculating the Mean and Variance of a Probability Distribution**
**Introduction:**
In this section, we will determine the mean and variance for a given probability distribution.
**Given Probability Distribution:**
The probability distribution is defined by the function:
\[ f(x) = \frac{2x}{k(k+1)}, \quad x = 1, 2, 3, \ldots, k \]
**Explanation:**
- \( f(x) \) represents the probability function for values \( x \) ranging from 1 to \( k \).
- The function is a probability distribution, where the probabilities are dependent on \( x \) and the constant \( k \).
**Objective:**
Our objective is to find the mean (expected value) and variance of this probability distribution.
**Steps to Understand:**
1. **Mean (Expected Value):**
- The mean is calculated using the formula:
\[ E(X) = \sum_{x=1}^{k} x \cdot f(x) \]
2. **Variance:**
- Variance is calculated as:
\[ Var(X) = E(X^2) - (E(X))^2 \]
- Where \( E(X^2) = \sum_{x=1}^{k} x^2 \cdot f(x) \)
In the next sections, we will derive these values using the given probability distribution function.
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