If `Sigma(x_(i)-2)=10,Sigma(y_(i)-5)=20,Sigmax_(i)y_(i)=148andn=5`, find cov `(x,y)`

Answer

To find the covariance \( \text{cov}(x, y) \), we will use the given information and the formula for covariance. Let's break down the steps:

### Step 1: Extract Given Information
We have the following data:
- \( \Sigma (x_i - 2) = 10 \)
- \( \Sigma (y_i - 5) = 20 \)
- \( \Sigma x_i y_i = 148 \)
- \( n = 5 \)

### Step 2: Calculate \( \Sigma x_i \)
From the equation \( \Sigma (x_i - 2) = 10 \), we can rewrite it as:
\[ \Sigma x_i - \Sigma 2 = 10 \]
Since \( \Sigma 2 = 2n \) (where \( n = 5 \)):
\[ \Sigma x_i - 10 = 10 \]
Thus, we have:
\[ \Sigma x_i = 20 \]

### Step 3: Calculate \( \Sigma y_i \)
From the equation \( \Sigma (y_i - 5) = 20 \), we can rewrite it as:
\[ \Sigma y_i - \Sigma 5 = 20 \]
Again, since \( \Sigma 5 = 5n \):
\[ \Sigma y_i - 25 = 20 \]
Thus, we have:
\[ \Sigma y_i = 45 \]

### Step 4: Use the Covariance Formula
The formula for covariance is given by:
\[ \text{cov}(x, y) = \frac{\Sigma x_i y_i}{n} - \frac{\Sigma x_i \cdot \Sigma y_i}{n^2} \]

### Step 5: Substitute Values into the Formula
Now substituting the values we have:
- \( \Sigma x_i y_i = 148 \)
- \( n = 5 \)
- \( \Sigma x_i = 20 \)
- \( \Sigma y_i = 45 \)

We can calculate:
\[ \text{cov}(x, y) = \frac{148}{5} - \frac{20 \cdot 45}{5^2} \]

### Step 6: Calculate Each Term
Calculating \( \frac{148}{5} \):
\[ \frac{148}{5} = 29.6 \]

Calculating \( \frac{20 \cdot 45}{5^2} \):
\[ \frac{20 \cdot 45}{25} = \frac{900}{25} = 36 \]

### Step 7: Final Calculation
Now substituting back into the covariance formula:
\[ \text{cov}(x, y) = 29.6 - 36 = -6.4 \]

### Final Answer
Thus, the covariance \( \text{cov}(x, y) \) is:
\[ \text{cov}(x, y) = -6.4 \]

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