Question

Which of the following functions represents a cumulative distribution function?

• A. \[F_1(x) = \begin{cases} 0, & \text{if } x < \frac{\pi}{4} \\\sin x, & \text{if } \frac{\pi}{4} \leq x < \frac{3\pi}{4} \\1, & \text{if } x \geq \frac{3\pi}{4} \end{cases}\]
• B. \[F_2(x) = \begin{cases} 0, & \text{if } x < 0 \\2 \sin x, & \text{if } 0 \leq x < \frac{\pi}{4} \\1, & \text{if } x \geq \frac{\pi}{4} \end{cases}\]
• C. \[F_3(x) = \begin{cases} 0, & \text{if } x < 0 \\x, & \text{if } 0 \leq x lt; \frac{1}{3} \\\frac{1}{3} x + \frac{1}{3}, & \text{if } \frac{1}{3} \leq x < \frac{1}{2} \\1, & \text{if } x \geq \frac{1}{2} \end{cases}\]
• D. \[F_4(x) = \begin{cases} 0, & \text{if } x < 0 \\\sqrt{2} \sin x, & \text{if } 0 \leq x < \frac{\pi}{4} \\1, & \text{if } x \geq \frac{\pi}{4} \end{cases}\]

Answer 1

The Correct Option is D

The correct option is (D): \[F_4(x) = \begin{cases} 0, & \text{if } x < 0 \\\sqrt{2} \sin x, & \text{if } 0 \leq x < \frac{\pi}{4} \\1, & \text{if } x \geq \frac{\pi}{4} \end{cases}\]