Randomness maximised

A function \( f(x) \) is defined as
\[ f(x) = \begin{cases} 1 & \quad \text{if } 0 \leq x \leq 1\\ 0 & \quad \text{ otherwise}\\ \end{cases} \]

Consider \( n \) random variables \( X_1 , X_2 , \dots \dots X_n \) such that \( f(x) \) is probability distribution function of each of the random variables i.e., \( \forall \text{ } X_i \text{ }; i = 1, 2, \dots , n \), \( f(x) \) is the probability distribution function.

Let \( E_n \) denote the \( n^{\text{th} } \) power of the expected value of \( \text{max} \{ X_1 , X_2 , \dots \dots X_n \} \).

As \(n\) grows larger, the expression \( E_n \) is found to converge to \( \mathcal{E} \).

Evaluate \( \displaystyle \lfloor 1000 \mathcal{E} \rfloor \).

Details and Assumptions:

  • It is to be noted that limit is evaluated for \( E_n \) after having obtained it for a general \(n\). It is not that the expected value of an arbitrarily large number of random variables is raised to that same arbitrarily large number to get \( \mathcal{E} \).
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