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Let $X_{1}, X_{2}$ and $X_{3}$ be independent random variables with uniform distribution over $[0, \theta]$. Consider the following statements.

  1. $E\left[\max \left\{X_{1}, X_{2}, X_{3}\right\}\right]=\frac{3}{4} \theta$.
  2. $E\left[\max \left\{X_{1}, X_{2}\right\}\right]-E\left[\max \left\{X_{2}, X_{3}\right\}\right]=0$.
  3. $E\left[X_{1}\right]=\theta / 2$
  4. $E\left[\max \left\{X_{1}, X_{2}\right\}\right]=\frac{2}{3} \theta$

Which of the above statements is/are $\text{TRUE}?$

  1. Only $\text{(i)}$
  2. Only $\text{(ii)}$
  3. Only $\text{(iii)}$
  4. Only $\text{(iv)}$
  5. All of $\text{(i) – (iv)}$
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