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Consider a random experiment where a spinner is rotated, and the continuous random variable \( X \) represents the angle spun, measured in degrees, over the interval \([0, 360]\).
  1. Determine the probability density function of \( X \).
  2. Calculate \( P(90 \leq X \leq 180) \).
  3. Calculate \( P(X \geq 60) \).
  4. Calculate the expected value \( E(X) \).
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  1. \( f(x) = \frac{1}{360} \)
  2. \(\begin{aligned}[t]P(90 \leq X \leq 180) &= \int_{90}^{180} \frac{1}{360} \, \mathrm{d}x \\ &= \left[ \frac{1}{360} x \right]_{90}^{180} \\ &= \frac{1}{4}\\\end{aligned}\)
  3. \(\begin{aligned}[t]P(X \geq 60) &= \int_{60}^{360} \frac{1}{360} \, \mathrm{d}x \\&= \left[ \frac{1}{360} x \right]_{60}^{360} \\&= \frac{5}{6}\end{aligned}\)
  4. \(\begin{aligned}[t]E(X) &= \frac{0+360}{2}\\&= 180 \\\end{aligned}\)
}