# 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 $$\mathcal{E}$$.
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