\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
The random variable \(X\) with values on \([1, 2]\) has density \( f(x) = \frac{2}{x^2} \).
Find \(V(X)\).\FieldText{1}{
  • \(\begin{aligned}[t]E(X) &= \int_{1}^{2} x \cdot \frac{2}{x^2} \, dx \\&= \int_{1}^{2} \frac{2}{x} \, dx \\&= 2 \left[ \ln x \right]_{1}^{2} \\&= 2 (\ln 2 - \ln 1) \\&= 2 \ln 2 \end{aligned}\)
  • \(\begin{aligned}[t]\int_{1}^{2} x^2 \cdot \frac{2}{x^2} \, dx &= \int_{1}^{2} 2 \, dx \\&= 2 \left[ x \right]_{1}^{2} \\&= 2 (2 - 1) \\&= 2 \end{aligned}\)
  • \(\begin{aligned}[t]V(X) &= \int_{1}^{2} x^2 \cdot f(x) \, dx - [E(X)]^2 \\&= 2 - (2 \ln 2)^2 \\&= 2 - 4 \ln^2 2 \end{aligned}\)
}