Let $\mathcal{X}$ be some set of independent random variables. Define the partially ordered set on $\mathcal{X}$ by $X_i \prec X_j$ if and only if $\mathcal{P}\left\{X_i \le X_j\right\} \ge \frac{1}{2}$. Are there known weak conditions on $\mathcal{X}$ such that this ordering is a total ordering? (I am guessing there are multiple answers to this question.)

The counterexample of non-transitive dice illustrates that not all $\mathcal{X}$ have total orderings. However, if, for example, $\mathcal{X}$ consisted of Gaussian random variables, a total ordering is established by comparing the means.

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Let $\mathcal{X}$ be some set of independent random variables. Define the ordering on $\mathcal{X}$ by $X_i \prec X_j$ if and only if $\mathcal{P}\left\{X_i \le X_j\right\} \ge \frac{1}{2}$. Are there known weak conditions on $\mathcal{X}$ such that this ordering is a total ordering? A partial ordering? (I am guessing there are multiple answers to this question.)

The counterexample of non-transitive dice illustrates that not all $\mathcal{X}$ have partial orderings, much less total orderings. However, if, for example, $\mathcal{X}$ consisted of Gaussian random variables, a total ordering is established by comparing the means.

${}\qquad{}$

Let $\mathcal{X}$ be some set of independent random variables. Define the partially ordered set on $\mathcal{X}$ by $X_i \prec X_j$ if and only if $\mathcal{P}\left\{X_i <= X_j\right\} \ge \frac{1}{2}$. Are there known weak conditions on $\mathcal{X}$ such that this ordering is a total ordering? (I am guessing there are multiple answers to this question.)

The counterexample of non-transitive dice illustrates that not all $\mathcal{X}$ have total orderings. However, if, for example, $\mathcal{X}$ consisted of Gaussian random variables, a total ordering is established by comparing the means.

Let $\mathcal{X}$ be some set of independent random variables. Define the partially ordered set on $\mathcal{X}$ by $X_i \prec X_j$ if and only if $\mathcal{P}\left\{X_i \le X_j\right\} \ge \frac{1}{2}$. Are there known weak conditions on $\mathcal{X}$ such that this ordering is a total ordering? (I am guessing there are multiple answers to this question.)

The counterexample of non-transitive dice illustrates that not all $\mathcal{X}$ have total orderings. However, if, for example, $\mathcal{X}$ consisted of Gaussian random variables, a total ordering is established by comparing the means.

${}\qquad{}$