If X and Y are random variables and a, b, c, d are constants such that `ane0, cne0` then
`r(aX+b,cY+d)` is

Options

A. `r(X,Y)`
B. `ac r(X,Y)`
C. `(|ac|)/(ac)r(X,Y)`
D. none of these

Answer

To solve the problem of finding \( r(aX + b, cY + d) \) where \( a, b, c, d \) are constants and \( a \neq 0, c \neq 0 \), we will use the properties of covariance and variance. Here are the steps to derive the solution:

### Step-by-Step Solution:

1. **Understanding the Correlation Formula**:
The correlation coefficient \( r(X, Y) \) is defined as:
\[ r(X, Y) = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}} \]

2. **Applying the Transformation**:
We need to find \( r(aX + b, cY + d) \). Using the properties of covariance:
\[ \text{Cov}(aX + b, cY + d) = a \cdot c \cdot \text{Cov}(X, Y) \]
The constants \( b \) and \( d \) do not affect the covariance.

3. **Calculating the Variance**:
The variance of a transformed variable is given by:
\[ \text{Var}(aX + b) = a^2 \cdot \text{Var}(X) \]
and
\[ \text{Var}(cY + d) = c^2 \cdot \text{Var}(Y) \]

4. **Substituting into the Correlation Formula**:
Now substituting these results back into the correlation formula:
\[ r(aX + b, cY + d) = \frac{\text{Cov}(aX + b, cY + d)}{\sqrt{\text{Var}(aX + b) \cdot \text{Var}(cY + d)}} \]
This becomes:
\[ r(aX + b, cY + d) = \frac{a \cdot c \cdot \text{Cov}(X, Y)}{\sqrt{(a^2 \cdot \text{Var}(X)) \cdot (c^2 \cdot \text{Var}(Y))}} \]

5. **Simplifying the Expression**:
The denominator simplifies to:
\[ \sqrt{a^2 \cdot c^2 \cdot \text{Var}(X) \cdot \text{Var}(Y)} = |a| \cdot |c| \cdot \sqrt{\text{Var}(X) \cdot \text{Var}(Y)} \]
Thus, we can rewrite the correlation as:
\[ r(aX + b, cY + d) = \frac{a \cdot c \cdot \text{Cov}(X, Y)}{|a| \cdot |c| \cdot \sqrt{\text{Var}(X) \cdot \text{Var}(Y)}} \]

6. **Final Result**:
This simplifies to:
\[ r(aX + b, cY + d) = \frac{a \cdot c}{|a| \cdot |c|} \cdot r(X, Y) \]
Since \( a \) and \( c \) are non-zero constants, we can express this as:
\[ r(aX + b, cY + d) = \text{sgn}(a) \cdot \text{sgn}(c) \cdot r(X, Y) \]
where \( \text{sgn}(x) \) is the sign function.

### Conclusion:
Thus, the final answer is:
\[ r(aX + b, cY + d) = \frac{a \cdot c}{|a| \cdot |c|} \cdot r(X, Y) \]