Question

Let the random variable \( X \) have uniform distribution on the interval \( \left( \frac{\pi}{6}, \frac{\pi}{2} \right) \). Then
\[ P(\cos X>\sin X) \] is

• A. \( \frac{2}{3} \)
• B. \( \frac{1}{2} \)
• C. \( \frac{1}{3} \)
• D. \( \frac{1}{4} \)

Hint:

For uniform distributions, probabilities are determined by the length of the relevant intervals.

Answer 1

The Correct Option is D

Step 1: Understanding the inequality.
We are given the random variable \( X \) with a uniform distribution on \( \left( \frac{\pi}{6}, \frac{\pi}{2} \right) \), and we need to find the probability that \( \cos X>\sin X \).
Step 2: Solving the inequality.
To solve \( \cos X>\sin X \), we first solve for \( X \) using the equation \( \cos X = \sin X \), which gives \( X = \frac{\pi}{4} \). Thus, \( \cos X>\sin X \) when \( X<\frac{\pi}{4} \).
Step 3: Calculating the probability.
The probability is the fraction of the interval \( \left( \frac{\pi}{6}, \frac{\pi}{2} \right) \) where \( X<\frac{\pi}{4} \). The length of the interval where \( X<\frac{\pi}{4} \) is \( \frac{\pi}{4} - \frac{\pi}{6} = \frac{\pi}{12} \), and the total length of the interval \( \left( \frac{\pi}{6}, \frac{\pi}{2} \right) \) is \( \frac{\pi}{2} - \frac{\pi}{6} = \frac{\pi}{3} \). Thus, the probability is: \[ P(\cos X>\sin X) = \frac{\frac{\pi}{12}}{\frac{\pi}{3}} = \frac{1}{4} \]