\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
The random variable \(X\) with values on \([0, 2]\) has density \(f(x) = \frac{x}{2}\).
Find \(V(X)\).\FieldText{1}{
  • \(\begin{aligned}[t]E(X) &= \int_{0}^{2} \frac{x^2}{2} \, dx \\&= \left[ \frac{x^3}{6} \right]_{0}^{2} \\&= \frac{4}{3} \\\end{aligned}\)
  • \(\begin{aligned}[t]\int_{0}^{2} x^2 \cdot \frac{x}{2} \, dx &= \int_{0}^{2} \frac{x^3}{2} \, dx \\&= \left[ \frac{x^4}{8} \right]_{0}^{2} \\&= 2\end{aligned}\)
  • \( \begin{aligned}[t]V(X) &= \int_{0}^{2} x^2 \cdot f(x) \, dx - [E(X)]^2 \\&= 2 - \left(\frac{4}{3}\right)^2 \\&= \frac{2}{9} \end{aligned}\)
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