Discussion
Suppose a distribution function of random variable X is :
\[ F(x) = \begin{cases} 0, & \text{if } x \lt 0, \\ x/2, & \text{if } 0 \le x \lt 1, \\ 1, & \text{if } x \ge 1. \end{cases} \]
Then, which of the following statements is not correct ?
Answer: B
The statement that is not correct is E(X) = 1/6.
F rises with slope 1/2 on [0, 1) to the value 1/2, then jumps to 1 at \(x = 1\). So X has density 1/2 on (0, 1) and an atom of mass 1/2 at 1, which makes D correct. Then
\[ \begin{aligned} E(X) &= \int_{0}^{1} \tfrac{x}{2}\,dx + 1 \cdot \tfrac{1}{2} \\ &= \tfrac{1}{4} + \tfrac{1}{2} = \tfrac{3}{4} \end{aligned} \]
so C is correct and B is not. A is also correct: \(P(-1 \lt X \le 1/2) = F(1/2) - F(-1) = 1/4 - 0 = 1/4\).
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