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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Statistics

NDA 2024 · Mathematics · Statistics

x_i 1 2 3 n f_i 1 2^ -1 2^ -2 2^ -(n-1) What is the mean of the...

x_i123\ldotsn
f_i12^{-1}2^{-2}\ldots2^{-(n-1)}
What is the mean of the distribution?

Options

  1. A2n+1−n+22n−1\frac{2^{n+1}-n+2}{2^n-1}
  2. B2n+1−n−22n−1\frac{2^{n+1}-n-2}{2^{n-1}}
  3. C2n+1−n−22n−1\frac{2^{n+1}-n-2}{2^n-1}
  4. D2n+1−n+22n\frac{2^{n+1}-n+2}{2^n}

Correct answer

C. 2n+1−n−22n−1\frac{2^{n+1}-n-2}{2^n-1}

Step-by-step solution

Σfi=1+12+122+…+12n−1Σfi=1−12n1−12=2n−12n−1\begin{aligned} & \Sigma f_i=1+\frac{1}{2}+\frac{1}{2^2}+\ldots+\frac{1}{2^{n-1}} \\ & \Sigma f_i=\frac{1-\frac{1}{2^n}}{1-\frac{1}{2}}=\frac{2^n-1}{2^{n-1}}\end{aligned} Now, Mean =ΣftxtΣft=2n+1−n−22n−1×2n−12n−1=2n+1−n−22n−1\begin{aligned} \text{ Now,} \quad \text{ Mean } & =\frac{\Sigma f_t x_t}{\Sigma f_t}=\frac{2^{n+1}-n-2}{2^{n-1}} \times \frac{2^{n-1}}{2^n-1} \\ & =\frac{2^{n+1}-n-2}{2^n-1} \end{aligned}$
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