Orthogonal transformation of standard normal sample
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I was reading this pdf https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec15.pdf
Shouldn't
$Var(Y_i)=sum_{k=1}^nv_{ki}^2$ (from how $Y_i$ is defined)
instead of
$$Var(Y_i)=sum_{k=1}^nv_{ik}^2$$ what author has written.
But if what author has written is correct, can you please explain why?
Also How $EY_iY_j=sum_{k=1}^nv_{ik}v_{jk}$?
Thanks.
statistics normal-distribution orthogonality
asked Dec 9 '18 at 15:19
q126yq126y
239212
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add a comment |
$begingroup$
I was reading this pdf https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec15.pdf
Shouldn't
$Var(Y_i)=sum_{k=1}^nv_{ki}^2$ (from how $Y_i$ is defined)
instead of
$$Var(Y_i)=sum_{k=1}^nv_{ik}^2$$ what author has written.
But if what author has written is correct, can you please explain why?
Also How $EY_iY_j=sum_{k=1}^nv_{ik}v_{jk}$?
Thanks.
statistics normal-distribution orthogonality
asked Dec 9 '18 at 15:19
q126yq126y
239212
$endgroup$
add a comment |
$begingroup$
I was reading this pdf https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec15.pdf
Shouldn't
$Var(Y_i)=sum_{k=1}^nv_{ki}^2$ (from how $Y_i$ is defined)
instead of
$$Var(Y_i)=sum_{k=1}^nv_{ik}^2$$ what author has written.
But if what author has written is correct, can you please explain why?
Also How $EY_iY_j=sum_{k=1}^nv_{ik}v_{jk}$?
Thanks.
statistics normal-distribution orthogonality
asked Dec 9 '18 at 15:19
q126yq126y
239212
$endgroup$
I was reading this pdf https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec15.pdf
Shouldn't
$Var(Y_i)=sum_{k=1}^nv_{ki}^2$ (from how $Y_i$ is defined)
instead of
$$Var(Y_i)=sum_{k=1}^nv_{ik}^2$$ what author has written.
But if what author has written is correct, can you please explain why?
Also How $EY_iY_j=sum_{k=1}^nv_{ik}v_{jk}$?
Thanks.
statistics normal-distribution orthogonality
statistics normal-distribution orthogonality
asked Dec 9 '18 at 15:19
q126yq126y
239212
asked Dec 9 '18 at 15:19
q126yq126y
239212
asked Dec 9 '18 at 15:19
q126yq126y
239212
asked Dec 9 '18 at 15:19
q126yq126y
239212
asked Dec 9 '18 at 15:19
q126yq126y
239212
239212
add a comment |
add a comment |
1 Answer
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Yes, technically $displaystyle{Var(Y_i) = sum_{k=1}^n v_{ki}^2}$ would be the more appropriate equation. But actually it doesn't really matter and $displaystyle{sum_{k=1}^n v_{ki}^2 = sum_{k=1}^n v_{ik}^2 = 1}$. This is because if $O$ is an orthogonal matrix, then $O^T$ is also an orthogonal matrix.
$$mathbb{E}[Y_i Y_j] = mathbb{E}left[ left(sum_{k=1}^n v_{ki} X_i right) left(sum_{k=1}^n v_{kj} X_i right) right]$$
If you expand that out, all the cross terms vanish because $mathbb{E}[X_i X_j] = 0$ for $i neq j$ (because $X_i$ and $X_j$ are uncorrelated).
answered Dec 9 '18 at 15:51
zoidbergzoidberg
1,065113
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$begingroup$
Thanks. Can you please also be so kind to tell how the moment generating function of iid X1,X2,...Xn was computed. it is on the start of page 2 of the pdf.
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– q126y
Dec 9 '18 at 16:06
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Perhaps you could post it as a separate question.
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– zoidberg
Dec 9 '18 at 16:06
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I have posted it here math.stackexchange.com/questions/3032570/… Thanks
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– q126y
Dec 9 '18 at 16:26
add a comment |
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1 Answer
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$begingroup$
Yes, technically $displaystyle{Var(Y_i) = sum_{k=1}^n v_{ki}^2}$ would be the more appropriate equation. But actually it doesn't really matter and $displaystyle{sum_{k=1}^n v_{ki}^2 = sum_{k=1}^n v_{ik}^2 = 1}$. This is because if $O$ is an orthogonal matrix, then $O^T$ is also an orthogonal matrix.
$$mathbb{E}[Y_i Y_j] = mathbb{E}left[ left(sum_{k=1}^n v_{ki} X_i right) left(sum_{k=1}^n v_{kj} X_i right) right]$$
If you expand that out, all the cross terms vanish because $mathbb{E}[X_i X_j] = 0$ for $i neq j$ (because $X_i$ and $X_j$ are uncorrelated).
answered Dec 9 '18 at 15:51
zoidbergzoidberg
1,065113
$endgroup$
$begingroup$
Thanks. Can you please also be so kind to tell how the moment generating function of iid X1,X2,...Xn was computed. it is on the start of page 2 of the pdf.
$endgroup$
– q126y
Dec 9 '18 at 16:06
$begingroup$
Perhaps you could post it as a separate question.
$endgroup$
– zoidberg
Dec 9 '18 at 16:06
$begingroup$
I have posted it here math.stackexchange.com/questions/3032570/… Thanks
$endgroup$
– q126y
Dec 9 '18 at 16:26
add a comment |
$begingroup$
Yes, technically $displaystyle{Var(Y_i) = sum_{k=1}^n v_{ki}^2}$ would be the more appropriate equation. But actually it doesn't really matter and $displaystyle{sum_{k=1}^n v_{ki}^2 = sum_{k=1}^n v_{ik}^2 = 1}$. This is because if $O$ is an orthogonal matrix, then $O^T$ is also an orthogonal matrix.
$$mathbb{E}[Y_i Y_j] = mathbb{E}left[ left(sum_{k=1}^n v_{ki} X_i right) left(sum_{k=1}^n v_{kj} X_i right) right]$$
If you expand that out, all the cross terms vanish because $mathbb{E}[X_i X_j] = 0$ for $i neq j$ (because $X_i$ and $X_j$ are uncorrelated).
answered Dec 9 '18 at 15:51
zoidbergzoidberg
1,065113
$endgroup$
$begingroup$
Thanks. Can you please also be so kind to tell how the moment generating function of iid X1,X2,...Xn was computed. it is on the start of page 2 of the pdf.
$endgroup$
– q126y
Dec 9 '18 at 16:06
$begingroup$
Perhaps you could post it as a separate question.
$endgroup$
– zoidberg
Dec 9 '18 at 16:06
$begingroup$
I have posted it here math.stackexchange.com/questions/3032570/… Thanks
$endgroup$
– q126y
Dec 9 '18 at 16:26
add a comment |
$begingroup$
Yes, technically $displaystyle{Var(Y_i) = sum_{k=1}^n v_{ki}^2}$ would be the more appropriate equation. But actually it doesn't really matter and $displaystyle{sum_{k=1}^n v_{ki}^2 = sum_{k=1}^n v_{ik}^2 = 1}$. This is because if $O$ is an orthogonal matrix, then $O^T$ is also an orthogonal matrix.
$$mathbb{E}[Y_i Y_j] = mathbb{E}left[ left(sum_{k=1}^n v_{ki} X_i right) left(sum_{k=1}^n v_{kj} X_i right) right]$$
If you expand that out, all the cross terms vanish because $mathbb{E}[X_i X_j] = 0$ for $i neq j$ (because $X_i$ and $X_j$ are uncorrelated).
answered Dec 9 '18 at 15:51
zoidbergzoidberg
1,065113
$endgroup$
Yes, technically $displaystyle{Var(Y_i) = sum_{k=1}^n v_{ki}^2}$ would be the more appropriate equation. But actually it doesn't really matter and $displaystyle{sum_{k=1}^n v_{ki}^2 = sum_{k=1}^n v_{ik}^2 = 1}$. This is because if $O$ is an orthogonal matrix, then $O^T$ is also an orthogonal matrix.
$$mathbb{E}[Y_i Y_j] = mathbb{E}left[ left(sum_{k=1}^n v_{ki} X_i right) left(sum_{k=1}^n v_{kj} X_i right) right]$$
If you expand that out, all the cross terms vanish because $mathbb{E}[X_i X_j] = 0$ for $i neq j$ (because $X_i$ and $X_j$ are uncorrelated).
answered Dec 9 '18 at 15:51
zoidbergzoidberg
1,065113
answered Dec 9 '18 at 15:51
zoidbergzoidberg
1,065113
answered Dec 9 '18 at 15:51
zoidbergzoidberg
1,065113
answered Dec 9 '18 at 15:51
zoidbergzoidberg
1,065113
1,065113
$begingroup$
Thanks. Can you please also be so kind to tell how the moment generating function of iid X1,X2,...Xn was computed. it is on the start of page 2 of the pdf.
$endgroup$
– q126y
Dec 9 '18 at 16:06
$begingroup$
Perhaps you could post it as a separate question.
$endgroup$
– zoidberg
Dec 9 '18 at 16:06
$begingroup$
I have posted it here math.stackexchange.com/questions/3032570/… Thanks
$endgroup$
– q126y
Dec 9 '18 at 16:26
add a comment |
$begingroup$
Thanks. Can you please also be so kind to tell how the moment generating function of iid X1,X2,...Xn was computed. it is on the start of page 2 of the pdf.
$endgroup$
– q126y
Dec 9 '18 at 16:06
$begingroup$
Perhaps you could post it as a separate question.
$endgroup$
– zoidberg
Dec 9 '18 at 16:06
$begingroup$
I have posted it here math.stackexchange.com/questions/3032570/… Thanks
$endgroup$
– q126y
Dec 9 '18 at 16:26
$begingroup$
Thanks. Can you please also be so kind to tell how the moment generating function of iid X1,X2,...Xn was computed. it is on the start of page 2 of the pdf.
$endgroup$
– q126y
Dec 9 '18 at 16:06
$begingroup$
Thanks. Can you please also be so kind to tell how the moment generating function of iid X1,X2,...Xn was computed. it is on the start of page 2 of the pdf.
$endgroup$
– q126y
Dec 9 '18 at 16:06
$begingroup$
Perhaps you could post it as a separate question.
$endgroup$
– zoidberg
Dec 9 '18 at 16:06
$begingroup$
Perhaps you could post it as a separate question.
$endgroup$
– zoidberg
Dec 9 '18 at 16:06
$begingroup$
I have posted it here math.stackexchange.com/questions/3032570/… Thanks
$endgroup$
– q126y
Dec 9 '18 at 16:26
$begingroup$
I have posted it here math.stackexchange.com/questions/3032570/… Thanks
$endgroup$
– q126y
Dec 9 '18 at 16:26
add a comment |
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