# On Realizable Distributions

Speaker:

## Time:

Saturday, 11 July 2020, 16:00 to 17:00

Join Zoom Meeting
https://zoom.us/j/98172222418?pwd=b0htSDQ5NDRKQ050K0d6cHJ3YnZXQT09
Meeting ID: 981 7222 2418
Given RVs X_1, X_2, X_3, X_4,...., X_n each taking a value in $\Omega$ and marginally distributed according to $\mu_1, \mu_2, ..., \mu_n$ on $\Omega^n$, we say a RV $Y$ distributed according to $\tau$ is realizable wrt to $X_1,..., X_n$ iff for all joint distributions $\mu$ of $X_1,.., X_n$ such that each $X_i~\mu_i$ there exists a joint distribution  $\mu'$ of $X_1,..., X_n, Y$ such that 1> $(X_1, X_2, ..., X_n)~\mu$  and $Y~\tau$ 2> Pr_{\mu'}[Y\in {X_1,..., X_n}]=1.
We will show that  a distribution $\tau$ is realizable wrt to $\mu_1,.., \mu_n$ iff for all $S\subseteq \Omega$ we have: $\mu_i(S)\leq \tau(S)\leq \mu_j (S)$ for some $i,j\in [n]$.