If ` u = (x-3)/( 2) and v= ( y-2)/( 3),` then cov(u,v) = k cov(x,y) . The value of k is

Options

A. `6`
B. ` (1)/(6)`
C. ` 5`
D. ` (1)/(5)`

Answer

To find the value of \( k \) in the equation \( \text{cov}(u,v) = k \cdot \text{cov}(x,y) \), where \( u = \frac{x-3}{2} \) and \( v = \frac{y-2}{3} \), we will follow these steps:

### Step 1: Express \( u \) and \( v \) in terms of \( x \) and \( y \)

Given:
\[ u = \frac{x - 3}{2} \]
\[ v = \frac{y - 2}{3} \]

### Step 2: Find the covariance \( \text{cov}(u,v) \)

The formula for covariance is:
\[ \text{cov}(u,v) = \text{E}[uv] - \text{E}[u]\text{E}[v] \]

### Step 3: Calculate \( \text{E}[u] \) and \( \text{E}[v] \)

Using the properties of expectation:
\[ \text{E}[u] = \text{E}\left[\frac{x - 3}{2}\right] = \frac{1}{2}(\text{E}[x] - 3) = \frac{\mu_x - 3}{2} \]
\[ \text{E}[v] = \text{E}\left[\frac{y - 2}{3}\right] = \frac{1}{3}(\text{E}[y] - 2) = \frac{\mu_y - 2}{3} \]

### Step 4: Calculate \( \text{E}[uv] \)

Now, we can express \( uv \):
\[ uv = \left(\frac{x - 3}{2}\right)\left(\frac{y - 2}{3}\right) = \frac{(x - 3)(y - 2)}{6} \]

Thus,
\[ \text{E}[uv] = \frac{1}{6} \text{E}[(x - 3)(y - 2)] \]

### Step 5: Expand \( \text{E}[(x - 3)(y - 2)] \)

Expanding gives:
\[ \text{E}[(x - 3)(y - 2)] = \text{E}[xy - 2x - 3y + 6] = \text{E}[xy] - 2\text{E}[x] - 3\text{E}[y] + 6 \]

### Step 6: Substitute back into covariance formula

Now substituting into the covariance formula:
\[ \text{cov}(u,v) = \frac{1}{6} \left(\text{E}[xy] - 2\mu_x - 3\mu_y + 6\right) - \left(\frac{\mu_x - 3}{2}\right)\left(\frac{\mu_y - 2}{3}\right) \]

### Step 7: Simplify the covariance expression

This gives:
\[ \text{cov}(u,v) = \frac{1}{6} \text{cov}(x,y) \]

### Step 8: Compare with the original equation

From the original equation \( \text{cov}(u,v) = k \cdot \text{cov}(x,y) \), we can see that:
\[ k = \frac{1}{6} \]

### Final Answer

Thus, the value of \( k \) is:
\[ \boxed{\frac{1}{6}} \]