The mean of the series `x_1, x_2,...x_n` is `barX`. If `x_2` is replaced by `lambda` then the new mean is

Options

A. `bar(X) + x_(2) + lambda`
B. `(bar(X)-x_(2)-lambda)/(n)`
C. `((n-1)bar(X)+lambda)/(n)`
D. `(n bar(X)-x_(2)+lambda)/(n)`

Answer

To find the new mean after replacing \( x_2 \) with \( \lambda \) in the series \( x_1, x_2, \ldots, x_n \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Original Mean**:
The mean of the original series is given by:
\[ \bar{X} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]
This means:
\[ x_1 + x_2 + x_3 + \ldots + x_n = n \bar{X} \]

**Hint**: Remember that the mean is the sum of all terms divided by the number of terms.

2. **Replacing \( x_2 \) with \( \lambda \)**:
When we replace \( x_2 \) with \( \lambda \), the new series becomes \( x_1, \lambda, x_3, \ldots, x_n \). The sum of the new series is:
\[ x_1 + \lambda + x_3 + \ldots + x_n \]

3. **Expressing the New Sum**:
We can express the new sum as:
\[ \text{New Sum} = (x_1 + x_2 + x_3 + \ldots + x_n) - x_2 + \lambda \]
Substituting the original sum:
\[ \text{New Sum} = n \bar{X} - x_2 + \lambda \]

**Hint**: When replacing a term in a sum, think about how to express the new sum in terms of the original sum.

4. **Calculating the New Mean**:
The new mean \( \bar{X}_{new} \) can be calculated as:
\[ \bar{X}_{new} = \frac{\text{New Sum}}{n} = \frac{n \bar{X} - x_2 + \lambda}{n} \]

5. **Simplifying the New Mean**:
We can simplify this expression:
\[ \bar{X}_{new} = \bar{X} + \frac{\lambda - x_2}{n} \]

**Hint**: When simplifying fractions, remember to distribute the division across the terms in the numerator.

### Final Result:
The new mean after replacing \( x_2 \) with \( \lambda \) is:
\[ \bar{X}_{new} = \bar{X} + \frac{\lambda - x_2}{n} \]