# #StackBounty: #probability #distributions #random-variable Existence of a random vector such that the differences of its components sat…

### Bounty: 50

Let us fix any three numbers in $$[0,1]$$ and summing up to $$1$$. I denote them by $$p_1, p_2, p_3$$.

Could you help to show that, for every possible vector of reals $$Uequiv (U_0, U_1, U_2)in mathbb{R}^3$$, there exists a random vector $$epsilonequiv (epsilon_0, epsilon_1, epsilon_2)$$ continuously distributed on $$mathbb{R}^3$$ such that the following equalities hold:
$$begin{cases} p_1=Pr(epsilon_1-epsilon_0geq U_0-U_1, epsilon_1-epsilon_2geq U_2-U_1)\ p_2=Pr(epsilon_2-epsilon_0geq U_0-U_2, epsilon_1-epsilon_2leq U_2-U_1)\ p_3=Pr(epsilon_1-epsilon_0leq U_0-U_1, epsilon_2-epsilon_0leq U_0-U_2) end{cases}$$

This question is related to a problem of identification in econometrics.

Following the comments below, I first reduce the dimension of my inequalities. In fact,
$$begin{cases} Pr(epsilon_1-epsilon_0geq U_0-U_1, epsilon_1-epsilon_2geq U_2-U_1)=Pr(eta_1geq -V_1, eta_1-eta_2geq V_2-V_1)\ Pr(epsilon_2-epsilon_0geq U_0-U_2, epsilon_1-epsilon_2leq U_2-U_1)=Pr(eta_2geq -V_2, eta_1-eta_2leq V_2-V_1)\ Pr(epsilon_1-epsilon_0leq U_0-U_1, epsilon_2-epsilon_0leq U_0-U_2)=Pr(eta_1leq -V_1, eta_2leq -V_2) end{cases}$$

where
$$eta_1equiv epsilon_1-epsilon_0\ eta_2equiv epsilon_2-epsilon_0\ V_1equiv U_1-U_0\ V_2equiv U_2-U_0\$$

Consider the regions
begin{aligned} &mathcal{R}{1,U}equiv {(eta_1,eta_2)in mathbb{R}^2: eta_1geq -V{1}, eta_1-eta_2geq V_{2}-V_{1}}\ & mathcal{R}{2,U}equiv {(eta_1,eta_2)in mathbb{R}^2: eta_2geq -V{2}, eta_1-eta_2leq V_{2}-V_{1}}\ & mathcal{R}{3,U}equiv {(eta_1,eta_2)in mathbb{R}^2: eta_1leq -V_1, eta_2leq -V_2}\ end{aligned}
These regions are non-empty and non-overlapping (except for the edges which, however, have zero probability measure). Further, they have a common vertex with coordinates $$(-V$$
{1},-V_{2})\$.

I now construct a continuous distribution for $$(eta_1, eta_2)$$ such that
$$begin{equation} label{eta_system} begin{cases} p_1=Pr(eta_1geq -V_1, eta_1-eta_2geq V_2-V_1)\ p_2=Pr(eta_2geq -V_2, eta_1-eta_2leq V_2-V_1)\ p_3=Pr(eta_1leq -V_1, eta_2leq -V_2) end{cases} end{equation}$$
Consider a bivariate normal distribution, $$mathcal{N}2(mu, Sigma{kappa_1,kappa_2})$$ with mean
$$muequiv (-V_1,-V_2)$$
and variance-covariance matrix
$$Sigma_{tau_1,tau_2}equiv begin{pmatrix} 5 & tau_1\ tau_1 & tau_2 end{pmatrix}$$
We can show that there exists values for $$(tau_1,tau_2)$$ such that system above is satisfied for $$etasim mathcal{N}2(mu, Sigma{tau_1,tau_2})$$ [HOW?].

Let $$epsilon_0sim mathcal{N}(0,1)$$. Let $$epsilon_1equiv eta_1+epsilon_0$$ and $$epsilon_2equiv eta_2+epsilon_0$$. These $$epsilon$$ satisfy my original system

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