\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Let \( X \) be a continuous random variable following a continuous uniform distribution on \([a, b]\).
Prove that for all \( c, d \in [a, b] \),$$P(c \leq X \leq d) = \frac{d - c}{b - a}.$$\FieldText{1}{Let \( c, d \in [a, b] \).$$\begin{aligned}[t]P(c \leq X \leq d) &= \int_{c}^{d} \frac{1}{b - a} \, \mathrm{d}x \\&= \left[ \frac{x}{b - a} \right]_{c}^{d} \\&= \frac{d - c}{b - a}.\end{aligned}$$}