https://file.aiquickdraw.com/mathbot/file/1725095502830016pogkw.jpg

The image presents a problem involving categorical variables \(X\) and \(Y\) and their association, as well as mathematical expressions involving expectations, variances, and covariances.

The problem statement reads:
"Categorical variables X and Y are associated with each other. This means that:
\[ \text{cov} (X,Y) = \sigma_{xy} \neq 0 \]

Based on the given information, which of the following statements is true?":

The statements to choose from are:

1. \(\text{E}[X + Y] = \text{E}[X] + \text{E}[Y]\)
2. \(\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)\)
3. \(\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y)\)

Options:

- a) Only [1] and [3]
- b) Only [1] and [2]
- c) Only [3]
- d) All statements are true.

### Analysis:

1. **Statement 1**: \(\text{E}[X + Y] = \text{E}[X] + \text{E}[Y]\)  
   - This statement is always true. The expectation of the sum of two random variables is equal to the sum of their expectations, regardless of whether they are associated.

2. **Statement 2**: \(\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)\)  
   - This statement is true only if \(X\) and \(Y\) are uncorrelated (i.e., \(\text{Cov}(X,Y) = 0\)). Since \(X\) and \(Y\) are associated and \(\sigma_{xy} \neq 0\), this statement is **not true**.

3. **Statement 3**: \(\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y)\)  
   - This is the correct formula for the variance of the sum of two random variables when they are associated. Since \(\text{Cov}(X,Y) \neq 0\), this statement is true.

### Conclusion:
The correct choice is **c) Only [3]**.

Would you like a detailed breakdown of each step or have any further questions?

### Related Questions:
1. How does covariance affect the variance of the sum of two random variables?
2. What are the conditions under which \(\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)\)?
3. Can you provide an example where \(\text{Var}(X + Y)\) differs from \(\text{Var}(X) + \text{Var}(Y)\)?
4. How would you compute the expectation of a non-linear function of two random variables?
5. What is the relationship between correlation and covariance?

### Tip:
Always remember that the variance of the sum of two random variables includes a term for their covariance, which can significantly impact the result if the variables are not independent.

Explore the relationship between covariance and variance in categorical variables X and Y. Learn about their association and how it affects mathematical expressions involving expectations, variances, and covariances. Suitable for advanced studies.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation