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Question #40621

Difficulty: 8/10 | 🟢 Free

The Question

A discrete random variable $X$ has a probability mass function given by $P(X = k) = \lambda \binom{6}{k} \left(\frac{1}{2}\right)^6 I(k \ge 2)$ for $k \in \{0, 1, 2, 3, 4, 5, 6\}$ , where $\lambda$ is a constant and $I(A)$ is the indicator function which equals $1$ if $A$ is true and $0$ otherwise. Find the value of $361 \text{ Var}(X)$ .

Options

A.
374
B.
160
C.
412
D.
122

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

General Term Formulation|Variance of Discrete Variable|Indicator Function Logic

Why This Question Is Tricky

Common Mistakes

This question tests the 'Boundary Condition Error' in the context of a truncated Binomial distribution.

The indicator function $I(k \ge 2)$ acts as a filter, removing the $k=0$ and $k=1$ terms from the sample space.

A common lethal mistake is for students to calculate the normalization constant $\lambda$ correctly, but then blindly apply the standard Binomial expectation formulas $E[X] = np$ and $E[X^2] = npq + n^2p^2$ multiplied by $\lambda$ .

This ignores the fact that the summation limits have changed, requiring the explicit subtraction of the $k=1$ term from the standard series sums $\sum k \binom{n}{k}$ and $\sum k^2 \binom{n}{k}$ .

Failing to subtract this boundary term leads directly to the trap answers (e.g., 160).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	General Term Formulation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free