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

Answer: (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 2

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