NDA 1 2024 Maths, question 118

Topic: Statistics

Passage · Q117 to Q118

$x_{i}$	$1$	$2$	$3$	$\ldots$	$n$
$f_{i}$	$1$	$2^{-1}$	$2^{-2}$	$\ldots$	$2^{-(n-1)}$

Q118

What is the mean of the distribution ?

(a) $\frac{2^{n+1}-n+2}{2^{n}-1}$
(b) $\frac{2^{n+1}-n-2}{2^{n-1}}$
(c) $\frac{2^{n+1}-n-2}{2^{n}-1}$ Correct answer
(d) $\frac{2^{n+1}-n+2}{2^{n}}$

Solution.

Answer: (c) $\frac{2^{n+1}-n-2}{2^{n}-1}$

The total frequency is 1 + 1/2 + ··· + 1/2^(n−1) = (2ⁿ − 1)/2^(n−1).
Divide the weighted sum [2^(n+1) − n − 2]/2^(n−1) by this total.
The mean is [2^(n+1) − n − 2]/(2ⁿ − 1).