Question:
If \(\bar{x}\) is the mean of \(x_1, x_2, \ldots, x_n\), then for a * 0, then the mean of \(ax_1, ax_2, \ldots, ax_n, \dfrac{x_1}{a}, \dfrac{x_2}{a}, \ldots, \dfrac{x_n}{a}\) is ________.
यदि \(\bar{x}\), \(x_1, x_2, \ldots, x_n\) का माध्य है, तो \(a \neq 0\) के लिए, \(ax_1, ax_2, \ldots, ax_n, \dfrac{x_1}{a}, \dfrac{x_2}{a}, \ldots, \dfrac{x_n}{a}\) का माध्य ________ है।
Options:
- \(\left(a + \dfrac{1}{a}\right)\dfrac{\bar{x}}{2}\)
- \(\dfrac{\left(a + \dfrac{1}{a}\right)\bar{x}}{2n}\)
- \(\left(a + \dfrac{1}{a}\right)\bar{x}\)
- \(\left(a + \dfrac{1}{a}\right)\dfrac{\bar{x}}{n}\)
- \(\left(a + \dfrac{1}{a}\right)\dfrac{\bar{x}}{2}\)
- \(\dfrac{\left(a + \dfrac{1}{a}\right)\bar{x}}{2n}\)
- \(\left(a + \dfrac{1}{a}\right)\bar{x}\)
- \(\left(a + \dfrac{1}{a}\right)\dfrac{\bar{x}}{n}\)
Correct Answer: a)
Explanation:
Sum of all \(2n\) terms \(= a\sum x_i + \dfrac{1}{a}\sum x_i = \left(a + \dfrac{1}{a}\right)n\bar{x}\).
Mean \(= \dfrac{(a + 1/a)n\bar{x}}{2n} = \left(a + \dfrac{1}{a}\right)\dfrac{\bar{x}}{2}\).
सभी \(2n\) पदों का योग \(= \left(a + \dfrac{1}{a}\right)n\bar{x}\)।
माध्य \(= \dfrac{(a + 1/a)n\bar{x}}{2n} = \left(a + \dfrac{1}{a}\right)\dfrac{\bar{x}}{2}\)।