Before You Begin

Question #40998

Difficulty: 8/10 | 🟢 Free

The Question

Consider a frequency distribution for a discrete variate $X$ , which takes values $x_r = 2|r - 10|$ with corresponding frequencies $f_r = \binom{20}{r}$ for $r = 0, 1, 2, \dots, 20$ . Let $P = \frac{1}{2^{20}} \binom{20}{10}$ . If the variance of $X$ is $V$ , then $V$ is equal to:

Options

A.
$20 - 400 P^2$
B.
$20$
C.
$20 - 100 P^2$
D.
$100 - 400 P^2$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Step-Deviation Method (u = (x-a)/h)|Coefficient of x^r in Expansion|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

The question tests the application of the Step-Deviation method on a Binomial frequency distribution, specifically injecting a 'Modulus Trap'.

Students often incorrectly assume that since the mean of $(r-10)$ is 0, the mean of $|r-10|$ is also 0, leading them to calculate the variance simply as $E(X^2) = 20$ (Option B).

The modulus forces a case-based partition of the summation, shifting the mean to $20P$ .

The variance must then be re-normalized by subtracting the square of this new mean, testing both algebraic stamina and conceptual clarity regarding absolute deviations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Step-Deviation Method (u = (x-a)/h)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free