# Randomness maximised

**Discrete Mathematics**Level 5

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 \( \mathcal{E} \).