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Question #41002

Difficulty: 8/10 | 🟢 Free

The Question

A student calculates the variance of a discrete random variable $X \in \{10, 20, 30\}$ by incorrectly assuming a uniform probability mass function. Later, it is found that the true PMF assigns equal probabilities to $X=10$ and $X=20$ , but a different probability to $X=30$ . If the true expected value of $X$ is $\frac{369}{15}$ , what is the true variance of $X$ ?

Options

A.
$\frac{1521}{25}$
B.
$\frac{1421}{25}$
C.
$\frac{1666}{25}$
D.
$\frac{1251}{25}$

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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

Correction of Incorrect Mean/Variance|Probability Mass Function (PMF)|Equiprobable Outcomes

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to reconstruct a Probability Mass Function (PMF) from a corrected mean while strictly enforcing the normalization constraint ( $\sum P(x_i) = 1$ ).

The narrative trap of the 'incorrect uniform assumption' tempts students to apply a proportional weight correction to the final outcome without re-normalizing the entire distribution, which would lead to a negative or invalid variance.

The secondary numerical trap uses the unsimplified fraction $\frac{369}{15}$ to distract from the simple linear relationship $E[X] = 30 - 30p$ , penalizing those who attempt to force the incorrect denominator of 3 into the new expected value calculation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Correction of Incorrect Mean/Variance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free