If means `barX,barY` of the variates X and Y are eah zero and `sigma_(X)^(2)=sigma_(Y)^(2)=1 and r=r_(XY)ne1`, the value of b such that `X+Y and X+bY` are uncorrelated is

Options

A. `b=0`
B. `b=1`
C. `b=-1`
D. none of these

Answer

To solve the problem, we need to find the value of \( b \) such that the variates \( X + Y \) and \( X + bY \) are uncorrelated. We know that the means \( \bar{X} \) and \( \bar{Y} \) are both 0, the variances \( \sigma_X^2 \) and \( \sigma_Y^2 \) are both 1, and the correlation coefficient \( r_{XY} \) is not equal to 1.

### Step-by-Step Solution:

1. **Understanding Uncorrelated Variables**:
Two variables are uncorrelated if their covariance is zero. Therefore, we need to find \( b \) such that:
\[ \text{Cov}(X + Y, X + bY) = 0 \]

2. **Using the Covariance Formula**:
We can use the covariance formula for linear combinations of variables:
\[ \text{Cov}(A, B) = \text{Cov}(aX + bY, cX + dY) = ac \text{Var}(X) + bd \text{Var}(Y) + ad \text{Cov}(X, Y) \]
For our case, let \( A = X + Y \) and \( B = X + bY \):
- Here, \( a = 1, b = 1, c = 1, d = b \).

3. **Calculating Covariance**:
Using the covariance formula:
\[ \text{Cov}(X + Y, X + bY) = \text{Cov}(X, X) + \text{Cov}(X, bY) + \text{Cov}(Y, X) + \text{Cov}(Y, bY) \]
This simplifies to:
\[ \text{Cov}(X + Y, X + bY) = \text{Var}(X) + b \text{Cov}(X, Y) + \text{Cov}(Y, X) + b \text{Var}(Y) \]

4. **Substituting Known Values**:
Since \( \text{Var}(X) = \text{Var}(Y) = 1 \) and \( \text{Cov}(X, Y) = r_{XY} \):
\[ \text{Cov}(X + Y, X + bY) = 1 + b r_{XY} + r_{XY} + b \]
This can be rewritten as:
\[ \text{Cov}(X + Y, X + bY) = 1 + (b + 1) r_{XY} \]

5. **Setting Covariance to Zero**:
For \( X + Y \) and \( X + bY \) to be uncorrelated:
\[ 1 + (b + 1) r_{XY} = 0 \]

6. **Solving for \( b \)**:
Rearranging gives:
\[ (b + 1) r_{XY} = -1 \]
Thus,
\[ b + 1 = -\frac{1}{r_{XY}} \]
Therefore,
\[ b = -1 - \frac{1}{r_{XY}} \]

7. **Conclusion**:
Since \( r_{XY} \) is not equal to 1, we can conclude that the only value of \( b \) that satisfies the condition for uncorrelation is:
\[ b = -1 \]

### Final Answer:
The value of \( b \) such that \( X + Y \) and \( X + bY \) are uncorrelated is \( b = -1 \).