Dirchlet vectors conditioned on inner product being equal

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Assuming that $X_1, X_2 in mathbbR^n$ are two Dirichlet vectors with parameters $alpha, beta in mathbbR^n$ i.e,
$$ X_1 sim operatornameDirichlet(alpha), X_2 sim operatornameDirichlet(beta)$$



For Dirichlet vectors,$E[X1] = fracalphaalpha^Tmathbb1,~ E[X2] = fracbetabeta^Tmathbb1 $.



Let $V in mathbbR^n$ be a random vector and we know that $V^TX_1 = V^TX_2$. Conditioned on this, what is the expectation of $X_1$ and $X_2$?



Any help is greatly appreciated. Thanks.




probability conditional-expectation orthogonality




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    Assuming that $X_1, X_2 in mathbbR^n$ are two Dirichlet vectors with parameters $alpha, beta in mathbbR^n$ i.e,
    $$ X_1 sim operatornameDirichlet(alpha), X_2 sim operatornameDirichlet(beta)$$



    For Dirichlet vectors,$E[X1] = fracalphaalpha^Tmathbb1,~ E[X2] = fracbetabeta^Tmathbb1 $.



    Let $V in mathbbR^n$ be a random vector and we know that $V^TX_1 = V^TX_2$. Conditioned on this, what is the expectation of $X_1$ and $X_2$?



    Any help is greatly appreciated. Thanks.




    probability conditional-expectation orthogonality




    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Assuming that $X_1, X_2 in mathbbR^n$ are two Dirichlet vectors with parameters $alpha, beta in mathbbR^n$ i.e,
      $$ X_1 sim operatornameDirichlet(alpha), X_2 sim operatornameDirichlet(beta)$$



      For Dirichlet vectors,$E[X1] = fracalphaalpha^Tmathbb1,~ E[X2] = fracbetabeta^Tmathbb1 $.



      Let $V in mathbbR^n$ be a random vector and we know that $V^TX_1 = V^TX_2$. Conditioned on this, what is the expectation of $X_1$ and $X_2$?



      Any help is greatly appreciated. Thanks.




      probability conditional-expectation orthogonality




      share|cite|improve this question













      Assuming that $X_1, X_2 in mathbbR^n$ are two Dirichlet vectors with parameters $alpha, beta in mathbbR^n$ i.e,
      $$ X_1 sim operatornameDirichlet(alpha), X_2 sim operatornameDirichlet(beta)$$



      For Dirichlet vectors,$E[X1] = fracalphaalpha^Tmathbb1,~ E[X2] = fracbetabeta^Tmathbb1 $.



      Let $V in mathbbR^n$ be a random vector and we know that $V^TX_1 = V^TX_2$. Conditioned on this, what is the expectation of $X_1$ and $X_2$?



      Any help is greatly appreciated. Thanks.





      probability conditional-expectation orthogonality





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Aug 2 at 17:40









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