Question:
If variable takes values \(0, 1, 2, 3, \dots, n\) with frequencies \(1, C(n, 1), C(n, 2), C(n, 3), \dots, C(n, n)\) respectively, then the arithmetic mean is:
यदि चर क्रमशः \(1, C(n, 1), C(n, 2), C(n, 3), \dots, C(n, n)\) आवृत्तियों के साथ \(0, 1, 2, 3, \dots, n\) मान लेता है, तो अंकगणितीय माध्य है:
Options:
- \(2n\)
- \(n + 1\)
- \(n\)
- \(n/2\)
- \(2n\)
- \(n + 1\)
- \(n\)
- \(n/2\)
Correct Answer: d)
Explanation:
The arithmetic mean \(\bar{x}\) is given by:
\[\bar{x} = \frac{\sum_{r=0}^n f_r x_r}{\sum_{r=0}^n f_r}\]
Here, the values are \(x_r = r\) and frequencies are \(f_r = {}^nC_r\).
Total frequency:
\[\sum_{r=0}^n {}^nC_r = 2^n\]
Sum of products:
\[\sum_{r=0}^n r \cdot {}^nC_r = n \sum_{r=1}^n {}^{n-1}C_{r-1} = n \cdot 2^{n-1}\]
Therefore, the arithmetic mean is:
\[\bar{x} = \frac{n \cdot 2^{n-1}}{2^n} = \frac{n}{2}\]
Hence, the correct option is (d).
अंकगणितीय माध्य \(\bar{x}\) का सूत्र है:
\[\bar{x} = \frac{\sum_{r=0}^n f_r x_r}{\sum_{r=0}^n f_r}\]
यहाँ मान \(x_r = r\) और आवृत्तियाँ \(f_r = {}^nC_r\) हैं।
कुल आवृत्ति:
\[\sum_{r=0}^n {}^nC_r = 2^n\]
गुणनफलों का योग:
\[\sum_{r=0}^n r \cdot {}^nC_r = n \sum_{r=1}^n {}^{n-1}C_{r-1} = n \cdot 2^{n-1}\]
अतः अंकगणितीय माध्य:
\[\bar{x} = \frac{n \cdot 2^{n-1}}{2^n} = \frac{n}{2}\]
अतः सही विकल्प (d) है।