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Consider a scenario where the continuous random variable \( X \) represents the waiting time at a bus stop, uniformly distributed over the interval \([0, 10]\) minutes.
  1. Determine the probability density function of \( X \).
  2. Calculate \( P(X \leq 8) \).
  3. Calculate the expected value \( E(X) \).
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  1. \( f(x) = \frac{1}{10 - 0} = \frac{1}{10} \) for \( 0 \leq x \leq 10 \)
  2. \(\begin{aligned}[t]P(X \leq 8) &= \int_{0}^{8} \frac{1}{10} \, \mathrm{d}x \\&= \left[ \frac{1}{10} x \right]_{0}^{8} \\&= \frac{8 - 0}{10} \\&= \frac{4}{5}\end{aligned}\)
  3. \(\begin{aligned}[t]E(X) &= \frac{0+10}{2}\\&= 5\\\end{aligned}\)
}