Find cov (x, y) for the following data :
`{:(x,3,4,5,6,7),(y,8,7,6,5,4):}`

Answer

To find the covariance \( \text{cov}(x, y) \) for the given data, we will follow these steps:

### Step 1: Create a table for the data
We have the values of \( x \) and \( y \):
- \( x: 3, 4, 5, 6, 7 \)
- \( y: 8, 7, 6, 5, 4 \)

We can organize this data into a table:

| \( x \) | \( y \) |
|---------|---------|
| 3 | 8 |
| 4 | 7 |
| 5 | 6 |
| 6 | 5 |
| 7 | 4 |

### Step 2: Calculate \( \sum x \), \( \sum y \), and \( \sum xy \)
Next, we will calculate the necessary summations.

- \( \sum x = 3 + 4 + 5 + 6 + 7 = 25 \)
- \( \sum y = 8 + 7 + 6 + 5 + 4 = 30 \)
- \( \sum xy = (3 \times 8) + (4 \times 7) + (5 \times 6) + (6 \times 5) + (7 \times 4) \)

Calculating \( \sum xy \):
- \( 3 \times 8 = 24 \)
- \( 4 \times 7 = 28 \)
- \( 5 \times 6 = 30 \)
- \( 6 \times 5 = 30 \)
- \( 7 \times 4 = 28 \)

Now, summing these products:
\[ \sum xy = 24 + 28 + 30 + 30 + 28 = 140 \]

### Step 3: Determine the number of observations \( n \)
The number of observations \( n \) is the count of pairs, which is 5.

### Step 4: Use the covariance formula
The formula for covariance is given by:
\[ \text{cov}(x, y) = \frac{1}{n} \left( \sum xy - \frac{1}{n} \sum x \cdot \sum y \right) \]

Substituting the values we calculated:
\[ \text{cov}(x, y) = \frac{1}{5} \left( 140 - \frac{1}{5} \cdot 25 \cdot 30 \right) \]

Calculating \( \frac{1}{5} \cdot 25 \cdot 30 \):
\[ \frac{1}{5} \cdot 25 \cdot 30 = 150 \]

Now substituting this back into the covariance formula:
\[ \text{cov}(x, y) = \frac{1}{5} \left( 140 - 150 \right) = \frac{1}{5} \cdot (-10) = -2 \]

### Final Answer
Thus, the covariance \( \text{cov}(x, y) \) is \( -2 \).

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