Let \(Y_{1}, \ldots, Y_{n}\) be independent and identically distributed random variables whose distribution on \(\mathbb{R}\) is atom-free, and let \(r_{i}=\operatorname{rank}\left(Y_{i}ight) /(n+1)\) be the normalized rank vector. The treatment assignment vector has conditionally independent Bernoulli components with parameter \(T_{i} \sim \operatorname{Ber}\left(r_{i}ight)\) given \(Y\).

Deduce the following:

1. the components of \(T\) are exchangeable and \(T_{i} \sim \operatorname{Ber}(1 / 2)\);

2. for each pair \(i eq j\) the covariance is \(\operatorname{cov}\left(T_{i}, T_{j}ight)=-1 /(12(n+1))\);

3. every symmetric function of \(\left(T_{1}, \ldots, T_{n}ight)\) is independent of the vector \(Y\);

4. the sample mean is symmetrically distributed with mean one half and variance

\[
\operatorname{var}\left(\bar{T}_{n}ight)=\frac{n+2}{6 n(n+1)}
\]

as opposed to \(1 /(4 n)\) for the independent and identically distributed Bernoulli setting.

Check these calculations for \(n=1\) and \(n=2\).