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Question #41091

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a discrete random variable taking values in the set $\{0, 1, 2, \dots, 5055\}$ . The probability mass function of $X$ is given by $P(X = r) = \frac{a_r}{6^{1011}}$ , where $a_r$ is the coefficient of $x^r$ in the expansion of $(1 + x + x^2 + x^3 + x^4 + x^5)^{1011}$ . Let $F(x) = P(X \le x)$ be the cumulative distribution function of $X$ . Find the value of $\sum_{r=0}^{5055} 2^{[2 F(r)]}$ where $[ \cdot ]$ denotes the greatest integer function.

Options

A.
7584
B.
7585
C.
7586
D.
7587

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

Random Variable Mapping|Coefficient of x^r in Expansion|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze the cumulative distribution function (CDF) of a symmetric multinomial expansion under the transformation of a step function.

The primary trap lies in the boundary conditions and the exact median of the distribution.

Students often correctly identify the symmetry $F(r) + F(M-1-r) = 1$ but incorrectly assume $[2F(r)]$ splits symmetrically.

They fail to account for the fact that $F(r)$ equals exactly $1/2$ at the median $r=2527$ , causing an asymmetric jump in the greatest integer function.

Furthermore, missing the exact value $F(5055)=1$ leads to an incorrect evaluation of the final term.

These discontinuities force off-by-one errors, perfectly mapping to the incorrect options provided.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Random Variable Mapping
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free