Q$1$ If the $n$ observations $\{x_1,...,x_n\}$ are distinct, then we can represent a bootstrap resample by an $n-$tuple $(m_1,...,m_n)$ of non$-$negative integers, where $m_i$ indicates the

number of repetitions of $x_i$ in the resample. Show that there are $([2n-1],[n])$ such $n-$

tuple. If the $n-$tuple $(m_1,...,m_n)$ is assigned multinomial probability $([n],[m_1,...,m_n])n^{-n},$

then further demonstrate that the most probable $n-$tuple is $(1,...,1).$ Apply Stirling's formula and show that

$([2n-1],[n])\frac{2^{2n-1}}{\sqrt[\phantom{2}]{n}}$ and $([n],[1,$dots,$1])(\frac{1}{n}{)}^n\sqrt[\phantom{2}]{2n}{e}^{-n}$

Finally, prove that a given $x_i$ appears in a bootstrap resample with approximate probability $1-e^{-1}$~~$0\mathrm{.}632$