Mean vector and covariance matrix












0












$begingroup$


I am given a home work for one subject, but my probability theory course is just started, so I dont have enough information. Could someone help me with that?
Given:
$$begin{equation}
p_underline x(x)=left{
begin{array}{@{}ll@{}}
frac1pi, & text{if} x^2_1+x^2_2 < 1 \
0, & text{otherwise}
end{array}right.
end{equation} $$
Find the mean and covariance matrix of the random vector of:
$$ underline y=begin{bmatrix}
1 & -1 \
0 & 2 \
end{bmatrix}underline x + begin{bmatrix} 2 \ 3 \ end{bmatrix}
$$
Marginal distribution, mean and variance is already determined.
Help me please with doing mean and covariance matrix. If this will be explained and possibly given a link to some resource it would be quite helpful for me.
Thanx






matrices probability-theory covariance means







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    0












    $begingroup$


    I am given a home work for one subject, but my probability theory course is just started, so I dont have enough information. Could someone help me with that?
    Given:
    $$begin{equation}
    p_underline x(x)=left{
    begin{array}{@{}ll@{}}
    frac1pi, & text{if} x^2_1+x^2_2 < 1 \
    0, & text{otherwise}
    end{array}right.
    end{equation} $$
    Find the mean and covariance matrix of the random vector of:
    $$ underline y=begin{bmatrix}
    1 & -1 \
    0 & 2 \
    end{bmatrix}underline x + begin{bmatrix} 2 \ 3 \ end{bmatrix}
    $$
    Marginal distribution, mean and variance is already determined.
    Help me please with doing mean and covariance matrix. If this will be explained and possibly given a link to some resource it would be quite helpful for me.
    Thanx






    matrices probability-theory covariance means







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      I am given a home work for one subject, but my probability theory course is just started, so I dont have enough information. Could someone help me with that?
      Given:
      $$begin{equation}
      p_underline x(x)=left{
      begin{array}{@{}ll@{}}
      frac1pi, & text{if} x^2_1+x^2_2 < 1 \
      0, & text{otherwise}
      end{array}right.
      end{equation} $$
      Find the mean and covariance matrix of the random vector of:
      $$ underline y=begin{bmatrix}
      1 & -1 \
      0 & 2 \
      end{bmatrix}underline x + begin{bmatrix} 2 \ 3 \ end{bmatrix}
      $$
      Marginal distribution, mean and variance is already determined.
      Help me please with doing mean and covariance matrix. If this will be explained and possibly given a link to some resource it would be quite helpful for me.
      Thanx






      matrices probability-theory covariance means







      share|cite|improve this question









      $endgroup$




      I am given a home work for one subject, but my probability theory course is just started, so I dont have enough information. Could someone help me with that?
      Given:
      $$begin{equation}
      p_underline x(x)=left{
      begin{array}{@{}ll@{}}
      frac1pi, & text{if} x^2_1+x^2_2 < 1 \
      0, & text{otherwise}
      end{array}right.
      end{equation} $$
      Find the mean and covariance matrix of the random vector of:
      $$ underline y=begin{bmatrix}
      1 & -1 \
      0 & 2 \
      end{bmatrix}underline x + begin{bmatrix} 2 \ 3 \ end{bmatrix}
      $$
      Marginal distribution, mean and variance is already determined.
      Help me please with doing mean and covariance matrix. If this will be explained and possibly given a link to some resource it would be quite helpful for me.
      Thanx






      matrices probability-theory covariance means




      matrices probability-theory covariance means






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      asked Jan 11 at 13:09









      Hillbilly JoeHillbilly Joe

      164




      164






















          1 Answer
          1






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          $begingroup$

          Hint: not a complete solution, since you've already done most of the work, given what you've stated. Write



          $$underline{y} = begin{bmatrix}
          y_1 \
          y_2
          end{bmatrix}, underline{x} = begin{bmatrix}
          x_1 \
          x_2
          end{bmatrix}text{.}$$
          Then we have
          $$begin{bmatrix}
          y_1 \
          y_2
          end{bmatrix} = begin{bmatrix}
          1 & -1 \
          0 & 2
          end{bmatrix}begin{bmatrix}
          x_1 \
          x_2
          end{bmatrix} + begin{bmatrix}
          2 \
          3
          end{bmatrix} = begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}text{.}$$
          So, what this problem boils down to is finding the mean and covariance matrix of
          $$begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}text{.}$$
          The expected value is simply
          $$mathbb{E}left[begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix} right] = begin{bmatrix}
          mathbb{E}[x_1 - x_2 + 2]\
          mathbb{E}[2x_2+3]
          end{bmatrix}$$
          and the covariance matrix is
          $$text{Cov}left(begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}right) = begin{bmatrix}
          text{Cov}left(x_1 - x_2 + 2, x_1 - x_2 + 2right) & text{Cov}left(x_1 - x_2 + 2, 2x_2+3right) \
          text{Cov}left(2x_2+3, x_1 - x_2 + 2right) & text{Cov}left(2x_2+3, 2x_2+3right)
          end{bmatrix}text{.}$$
          I will let you handle it from here.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you for your help
            $endgroup$
            – Hillbilly Joe
            Jan 11 at 13:29











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          1 Answer
          1






          active

          oldest

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          1 Answer
          1






          active

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          active

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          active

          oldest

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          0












          $begingroup$

          Hint: not a complete solution, since you've already done most of the work, given what you've stated. Write



          $$underline{y} = begin{bmatrix}
          y_1 \
          y_2
          end{bmatrix}, underline{x} = begin{bmatrix}
          x_1 \
          x_2
          end{bmatrix}text{.}$$
          Then we have
          $$begin{bmatrix}
          y_1 \
          y_2
          end{bmatrix} = begin{bmatrix}
          1 & -1 \
          0 & 2
          end{bmatrix}begin{bmatrix}
          x_1 \
          x_2
          end{bmatrix} + begin{bmatrix}
          2 \
          3
          end{bmatrix} = begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}text{.}$$
          So, what this problem boils down to is finding the mean and covariance matrix of
          $$begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}text{.}$$
          The expected value is simply
          $$mathbb{E}left[begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix} right] = begin{bmatrix}
          mathbb{E}[x_1 - x_2 + 2]\
          mathbb{E}[2x_2+3]
          end{bmatrix}$$
          and the covariance matrix is
          $$text{Cov}left(begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}right) = begin{bmatrix}
          text{Cov}left(x_1 - x_2 + 2, x_1 - x_2 + 2right) & text{Cov}left(x_1 - x_2 + 2, 2x_2+3right) \
          text{Cov}left(2x_2+3, x_1 - x_2 + 2right) & text{Cov}left(2x_2+3, 2x_2+3right)
          end{bmatrix}text{.}$$
          I will let you handle it from here.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you for your help
            $endgroup$
            – Hillbilly Joe
            Jan 11 at 13:29
















          0












          $begingroup$

          Hint: not a complete solution, since you've already done most of the work, given what you've stated. Write



          $$underline{y} = begin{bmatrix}
          y_1 \
          y_2
          end{bmatrix}, underline{x} = begin{bmatrix}
          x_1 \
          x_2
          end{bmatrix}text{.}$$
          Then we have
          $$begin{bmatrix}
          y_1 \
          y_2
          end{bmatrix} = begin{bmatrix}
          1 & -1 \
          0 & 2
          end{bmatrix}begin{bmatrix}
          x_1 \
          x_2
          end{bmatrix} + begin{bmatrix}
          2 \
          3
          end{bmatrix} = begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}text{.}$$
          So, what this problem boils down to is finding the mean and covariance matrix of
          $$begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}text{.}$$
          The expected value is simply
          $$mathbb{E}left[begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix} right] = begin{bmatrix}
          mathbb{E}[x_1 - x_2 + 2]\
          mathbb{E}[2x_2+3]
          end{bmatrix}$$
          and the covariance matrix is
          $$text{Cov}left(begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}right) = begin{bmatrix}
          text{Cov}left(x_1 - x_2 + 2, x_1 - x_2 + 2right) & text{Cov}left(x_1 - x_2 + 2, 2x_2+3right) \
          text{Cov}left(2x_2+3, x_1 - x_2 + 2right) & text{Cov}left(2x_2+3, 2x_2+3right)
          end{bmatrix}text{.}$$
          I will let you handle it from here.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you for your help
            $endgroup$
            – Hillbilly Joe
            Jan 11 at 13:29














          0












          0








          0





          $begingroup$

          Hint: not a complete solution, since you've already done most of the work, given what you've stated. Write



          $$underline{y} = begin{bmatrix}
          y_1 \
          y_2
          end{bmatrix}, underline{x} = begin{bmatrix}
          x_1 \
          x_2
          end{bmatrix}text{.}$$
          Then we have
          $$begin{bmatrix}
          y_1 \
          y_2
          end{bmatrix} = begin{bmatrix}
          1 & -1 \
          0 & 2
          end{bmatrix}begin{bmatrix}
          x_1 \
          x_2
          end{bmatrix} + begin{bmatrix}
          2 \
          3
          end{bmatrix} = begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}text{.}$$
          So, what this problem boils down to is finding the mean and covariance matrix of
          $$begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}text{.}$$
          The expected value is simply
          $$mathbb{E}left[begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix} right] = begin{bmatrix}
          mathbb{E}[x_1 - x_2 + 2]\
          mathbb{E}[2x_2+3]
          end{bmatrix}$$
          and the covariance matrix is
          $$text{Cov}left(begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}right) = begin{bmatrix}
          text{Cov}left(x_1 - x_2 + 2, x_1 - x_2 + 2right) & text{Cov}left(x_1 - x_2 + 2, 2x_2+3right) \
          text{Cov}left(2x_2+3, x_1 - x_2 + 2right) & text{Cov}left(2x_2+3, 2x_2+3right)
          end{bmatrix}text{.}$$
          I will let you handle it from here.






          share|cite|improve this answer









          $endgroup$



          Hint: not a complete solution, since you've already done most of the work, given what you've stated. Write



          $$underline{y} = begin{bmatrix}
          y_1 \
          y_2
          end{bmatrix}, underline{x} = begin{bmatrix}
          x_1 \
          x_2
          end{bmatrix}text{.}$$
          Then we have
          $$begin{bmatrix}
          y_1 \
          y_2
          end{bmatrix} = begin{bmatrix}
          1 & -1 \
          0 & 2
          end{bmatrix}begin{bmatrix}
          x_1 \
          x_2
          end{bmatrix} + begin{bmatrix}
          2 \
          3
          end{bmatrix} = begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}text{.}$$
          So, what this problem boils down to is finding the mean and covariance matrix of
          $$begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}text{.}$$
          The expected value is simply
          $$mathbb{E}left[begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix} right] = begin{bmatrix}
          mathbb{E}[x_1 - x_2 + 2]\
          mathbb{E}[2x_2+3]
          end{bmatrix}$$
          and the covariance matrix is
          $$text{Cov}left(begin{bmatrix}
          x_1 - x_2 + 2\
          2x_2+3
          end{bmatrix}right) = begin{bmatrix}
          text{Cov}left(x_1 - x_2 + 2, x_1 - x_2 + 2right) & text{Cov}left(x_1 - x_2 + 2, 2x_2+3right) \
          text{Cov}left(2x_2+3, x_1 - x_2 + 2right) & text{Cov}left(2x_2+3, 2x_2+3right)
          end{bmatrix}text{.}$$
          I will let you handle it from here.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 11 at 13:24









          ClarinetistClarinetist

          11k42778




          11k42778












          • $begingroup$
            Thank you for your help
            $endgroup$
            – Hillbilly Joe
            Jan 11 at 13:29


















          • $begingroup$
            Thank you for your help
            $endgroup$
            – Hillbilly Joe
            Jan 11 at 13:29
















          $begingroup$
          Thank you for your help
          $endgroup$
          – Hillbilly Joe
          Jan 11 at 13:29




          $begingroup$
          Thank you for your help
          $endgroup$
          – Hillbilly Joe
          Jan 11 at 13:29


















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