#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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