For observations `x_i` given `sum_(i=1)^10 (X_i - 5) = 10` and `sum_(i=1)^10 (X_i - 5)^2 = 40` If mean and variance of observations `(x_1 - 3),(x_2 - 3),(x_3 - 3).........(x_10 - 3)` is `lambda` & `mu` respectively then ordered pair `(lambda, mu)` is

Options

A. `(3,3)
B. `(1,3)`
C. `(3,1)`
D. `(1,1)

Explanation

To solve the problem, we need to find the mean and variance of the observations \( (x_i - 3) \) given the conditions provided.

### Step 1: Understand the given conditions
We have:
1. \( \sum_{i=1}^{10} (x_i - 5) = 10 \)
2. \( \sum_{i=1}^{10} (x_i - 5)^2 = 40 \)

### Step 2: Calculate the mean of \( x_i - 5 \)
The mean of the observations \( x_i - 5 \) can be calculated as follows:
\[ \text{Mean} = \frac{\sum_{i=1}^{10} (x_i - 5)}{10} = \frac{10}{10} = 1 \]
Thus, we denote this mean as \( \bar{x} - 5 = 1 \).

### Step 3: Calculate the variance of \( x_i - 5 \)
The variance is calculated using the formula:
\[ \text{Variance} = \frac{\sum_{i=1}^{10} (x_i - 5)^2}{10} - \left(\text{Mean}\right)^2 \]
Substituting the values we have:
\[ \text{Variance} = \frac{40}{10} - 1^2 = 4 - 1 = 3 \]

### Step 4: Calculate the mean of \( x_i - 3 \)
Using the property of means:
\[ \text{Mean}(x_i - 3) = \text{Mean}(x_i - 5) + 2 \]
Since we found the mean of \( x_i - 5 \) to be 1:
\[ \text{Mean}(x_i - 3) = 1 + 2 = 3 \]
Thus, we denote this mean as \( \lambda = 3 \).

### Step 5: Calculate the variance of \( x_i - 3 \)
Using the property of variance:
\[ \text{Variance}(x_i - 3) = \text{Variance}(x_i - 5) \]
Since we found the variance of \( x_i - 5 \) to be 3:
\[ \text{Variance}(x_i - 3) = 3 \]
Thus, we denote this variance as \( \mu = 3 \).

### Conclusion
The ordered pair \( (\lambda, \mu) \) is:
\[ (\lambda, \mu) = (3, 3) \]