7 The discrete random variable X has probability distribution given by (4x +7) P(X= x) = for x= 1, 2, 3, 4. 68 (i) Find (a) E(X) (b) E(X²) (c) E(X²+5X– 2). (ii) Verify that E(X² + 5X– 2) = E(X²) + 5E(X) – 2.
7 The discrete random variable X has probability distribution given by (4x +7) P(X= x) = for x= 1, 2, 3, 4. 68 (i) Find (a) E(X) (b) E(X²) (c) E(X²+5X– 2). (ii) Verify that E(X² + 5X– 2) = E(X²) + 5E(X) – 2.
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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![## Discrete Random Variable Probability Distribution
### Problem Statement
The discrete random variable \(X\) has a probability distribution given by:
\[ P(X = x) = \frac{4x + 7}{68} \quad \text{for} \; x = 1, 2, 3, 4. \]
### Questions
1. **Find:**
1. **(a)** \( E(X) \)
2. **(b)** \( E(X^2) \)
3. **(c)** \( E(X^2 + 5X - 2) \)
2. **Verify** that:
\[ E(X^2 + 5X - 2) = E(X^2) + 5E(X) - 2. \]
### Explanation
1. **Expected Value (\( E(X) \)):**
The expected value, \( E(X) \), of a discrete random variable is the sum of the product of each possible value of the variable and its corresponding probability.
2. **Expected Value of \( X^2 \) ( \( E(X^2) \)):**
The expected value of \( X^2 \) is calculated by summing the products of each possible value squared, and its corresponding probability.
3. **Expected Value of the Expression ( \( E(X^2 + 5X - 2) \)):**
For a linear combination of expected values, the expected value of the combined expression can be separated into the sum of the expected values of each term, each appropriately weighted.
### Verifying the Formula:
To verify \( E(X^2 + 5X - 2) = E(X^2) + 5E(X) - 2 \), we can use the properties of expected values, which states that:
\[ E(aX + bY + c) = aE(X) + bE(Y) + c \]
where \(a\), \(b\), and \(c\) are constants and \(X\) and \(Y\) are random variables. This property can be used to simplify and verify the given expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F732159e0-135e-441a-8a8d-d8e4bb926137%2F3c93a51b-9fa4-4339-931a-00e729b9a4d7%2Fbdt63we_processed.png&w=3840&q=75)
Transcribed Image Text:## Discrete Random Variable Probability Distribution
### Problem Statement
The discrete random variable \(X\) has a probability distribution given by:
\[ P(X = x) = \frac{4x + 7}{68} \quad \text{for} \; x = 1, 2, 3, 4. \]
### Questions
1. **Find:**
1. **(a)** \( E(X) \)
2. **(b)** \( E(X^2) \)
3. **(c)** \( E(X^2 + 5X - 2) \)
2. **Verify** that:
\[ E(X^2 + 5X - 2) = E(X^2) + 5E(X) - 2. \]
### Explanation
1. **Expected Value (\( E(X) \)):**
The expected value, \( E(X) \), of a discrete random variable is the sum of the product of each possible value of the variable and its corresponding probability.
2. **Expected Value of \( X^2 \) ( \( E(X^2) \)):**
The expected value of \( X^2 \) is calculated by summing the products of each possible value squared, and its corresponding probability.
3. **Expected Value of the Expression ( \( E(X^2 + 5X - 2) \)):**
For a linear combination of expected values, the expected value of the combined expression can be separated into the sum of the expected values of each term, each appropriately weighted.
### Verifying the Formula:
To verify \( E(X^2 + 5X - 2) = E(X^2) + 5E(X) - 2 \), we can use the properties of expected values, which states that:
\[ E(aX + bY + c) = aE(X) + bE(Y) + c \]
where \(a\), \(b\), and \(c\) are constants and \(X\) and \(Y\) are random variables. This property can be used to simplify and verify the given expression.
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