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.