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Question #41825

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_0^x (t^2 + t)[t] \, dt$ , where $[ \cdot ]$ denotes the greatest integer function. A discrete random variable $X$ takes values in the set $\{1, 2, 3, 4\}$ . The probability mass function of $X$ is given by: $P(X=k) = \lambda \left( f'_+(k) - f'_-(k) \right)$ where $f'_+(k)$ and $f'_-(k)$ denote the right-hand and left-hand derivatives of $f(x)$ at $x=k$ , respectively, and $\lambda$ is a positive constant. Find the expected value $E(X)$ .

Options

A.
$\frac{13}{4}$
B.
$\frac{15}{4}$
C.
$\frac{11}{3}$
D.
$\frac{7}{2}$

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

Probability Mass Function (PMF)|Newton-Leibniz Formula with variable limits|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the interaction between the Newton-Leibniz formula and step function discontinuities (Greatest Integer Function).

Students often incorrectly assume that the derivative of an integral with a variable upper limit is simply the integrand evaluated at that limit, ignoring points of discontinuity.

By defining the PMF through the difference of left and right derivatives at integer points, the problem forces the student to evaluate the jump discontinuity of the integrand.

The 'lethal trap' lies in correctly evaluating the left-hand limit of the GIF as $k-1$ versus the right-hand limit as $k$ , which directly dictates the probability distribution.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Probability Mass Function (PMF)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free