\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
The random variable \(X\) has the density \( f(x) = \frac{x}{2} \), on the interval \([0, 2]\).
Find \( P\left(1 \leq X \leq 2\right) \).
\FieldText{1}{$$\begin{aligned}[t] P\left(1 \leq X \leq 2\right) &= \int_{1}^{2} \frac{x}{2} \; \mathrm{d}x \\ &= \frac{1}{2} \int_{1}^{2} x \; \mathrm{d}x \\ &= \frac{1}{2} \left[ \frac{x^2}{2} \right]_{1}^{2} \\ &= \frac{1}{2} \left( \frac{2^2}{2} - \frac{1^2}{2} \right) \\ &= \frac{1}{2} \left( 2 - \frac{1}{2} \right) \\ &= \frac{1}{2} \cdot \frac{3}{2} \\ &= \frac{3}{4} \end{aligned}$$}