Let X be a random variable with distribution function \( F_X(x) = \begin{cases} 0 & \text{for } x < 0 \\ \frac{x}{8} & \text{for } 0 \le x < 1 \\ \frac{1}{4} + \frac{x}{8} & \text{for } 1 \le x < 2 \\ \frac{3}{4} + \frac{x}{12} & \text{for } 2 \le x < 3 \\ 1 & \text{for } x \ge 3 \end{cases} \) then \( P(1 \le X \le 2) \) is
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In CDF problems, always check if the function is continuous at the boundaries. If $F(a) \neq \lim F(x \to a^-)$, there is a discrete mass $P(X=a)$. For this variable, $P(X=1) = 5/8 - 1/4 = 3/8$ and $P(X=2) = 3/4 - 6/8 = 0$. Knowing these jumps is key.