If `u=(x-a)/(c) and v= (y-b)/ d` then `b_(xy)=`

Answer

To find \( b_{xy} \) given the transformations \( u = \frac{x - a}{c} \) and \( v = \frac{y - b}{d} \), we can follow these steps:

### Step 1: Understand the transformations
We have:
\[ u = \frac{x - a}{c} \quad \text{and} \quad v = \frac{y - b}{d} \]
From these equations, we can express \( x \) and \( y \) in terms of \( u \) and \( v \):
\[ x = a + cu \quad \text{and} \quad y = b + dv \]

### Step 2: Identify the relationship between \( b_{xy} \) and \( b_{uv} \)
The formula for the slope of the regression line (or the coefficient of correlation) can be expressed as:
\[ b_{xy} = r_{xy} \frac{\sigma_x}{\sigma_y} \]
and for \( u \) and \( v \):
\[ b_{uv} = r_{uv} \frac{\sigma_u}{\sigma_v} \]

### Step 3: Relate the standard deviations
From the transformations, we know:
\[ \sigma_x = c \sigma_u \quad \text{and} \quad \sigma_y = d \sigma_v \]

### Step 4: Substitute into the equations
Substituting the expressions for \( \sigma_x \) and \( \sigma_y \) into the equation for \( b_{xy} \):
\[ b_{xy} = r_{xy} \frac{c \sigma_u}{d \sigma_v} \]

### Step 5: Relate the correlation coefficients
Since the correlation coefficients are independent of the origin and scale, we can express:
\[ r_{xy} = r_{uv} \]

### Step 6: Final expression for \( b_{xy} \)
Now substituting \( r_{xy} \) and simplifying:
\[ b_{xy} = b_{uv} \frac{c}{d} \]

### Conclusion
Thus, the required expression for \( b_{xy} \) is:
\[ b_{xy} = \frac{c}{d} b_{uv} \]