If `Z=aX+bY` and r is the correlation coefficient between X and Y, then `sigma_(Z)^(2)` is equal to

Options

A. `2ab, rsigma_(X)sigma_(Y)`
B. `a^(2)sigma_(X)^(2)+b^(2)sigma_(Y)^(2)-2brsigma_(X)sigma_(Y)`
C. `a^(2)sigma_(X)^(2)+b^(2)sigma_(Y)^(2)+2abrsigma_(X)sigma_(Y)`
D. none of these

Answer

To solve the problem, we need to find the variance of the linear combination \( Z = aX + bY \), where \( r \) is the correlation coefficient between \( X \) and \( Y \).

### Step-by-Step Solution:

1. **Understand the Variance of a Linear Combination**:
The variance of a linear combination of two random variables \( X \) and \( Y \) is given by the formula:
\[ \sigma_Z^2 = a^2 \sigma_X^2 + b^2 \sigma_Y^2 + 2ab \text{Cov}(X, Y) \]
where \( \sigma_Z^2 \) is the variance of \( Z \), \( \sigma_X^2 \) is the variance of \( X \), \( \sigma_Y^2 \) is the variance of \( Y \), and \( \text{Cov}(X, Y) \) is the covariance between \( X \) and \( Y \).

2. **Relate Covariance to Correlation**:
The covariance can be expressed in terms of the correlation coefficient \( r \):
\[ \text{Cov}(X, Y) = r \sigma_X \sigma_Y \]
where \( r \) is the correlation coefficient between \( X \) and \( Y \).

3. **Substitute Covariance into the Variance Formula**:
Substitute the expression for covariance into the variance formula:
\[ \sigma_Z^2 = a^2 \sigma_X^2 + b^2 \sigma_Y^2 + 2ab (r \sigma_X \sigma_Y) \]

4. **Simplify the Expression**:
The expression now becomes:
\[ \sigma_Z^2 = a^2 \sigma_X^2 + b^2 \sigma_Y^2 + 2abr \sigma_X \sigma_Y \]

5. **Final Result**:
Therefore, the variance of \( Z \) is:
\[ \sigma_Z^2 = a^2 \sigma_X^2 + b^2 \sigma_Y^2 + 2abr \sigma_X \sigma_Y \]

### Conclusion:
The correct answer corresponds to the third option:
\[ \sigma_Z^2 = a^2 \sigma_X^2 + b^2 \sigma_Y^2 + 2abr \sigma_X \sigma_Y \]