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