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Question #41093

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a continuous random variable with the probability density function defined as: $f(x) = \begin{cases} k \left( \sin x + \frac{\sec^2 x}{1 + 2^{\tan x}} \right) & \text{for } x \in [0, \pi] \setminus \left\{\frac{\pi}{2}\right\} \\ 0 & \text{otherwise} \end{cases}$ If $f(x)$ represents a valid probability density function, what is the value of the constant $k$ ?

Options

A.
$\frac{1}{2}$
B.
$\frac{1}{4}$
C.
1
D.
No such real value of $k$ exists

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|Definite Integral Properties (King's Property)|Limit Existence at Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous evaluation of improper integrals and the definition of a valid probability density function.

The core trap lies in the blind application of King's Property ( $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$ ) and the Fundamental Theorem of Calculus across a non-removable discontinuity at $x = \pi/2$ .

Students who ignore the vertical asymptote of $\sec^2 x$ will incorrectly evaluate the integral as $0$ and deduce $k = 1/2$ .

However, splitting the integral at the discontinuity reveals that the area under the curve diverges to infinity, making it impossible to normalize the function to $1$ .

This highlights the necessity of checking continuity before applying standard integration properties.

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