Calculating variance of given formula [closed]
$begingroup$
Let $D_i$ be indicators, $iin{1, 2 ... k}$.
I am interested in calculating variance of $$Y = frac {sum_{i=1}^k x_iD_i}{sum_{i=1}^k n_iD_i}$$ where $x_i$ and $n_i$ are given real numbers, $i in {1, 2 ... k}$.
Exactly m of k indicators is equal to 1 and all possible combinations are equally possible.
Also, probability for each combination of choosing these m indicators is $frac {1}{k choose m}$
probability statistics variance
asked Dec 25 '18 at 9:24
dkutlesicdkutlesic
11
$endgroup$
closed as off-topic by Kavi Rama Murthy, Lee David Chung Lin, StubbornAtom, NCh, Namaste Dec 29 '18 at 21:24
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Kavi Rama Murthy, Lee David Chung Lin, StubbornAtom, NCh, Namaste
If this question can be reworded to fit the rules in the help center, please edit the question.
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$begingroup$
Let $D_i$ be indicators, $iin{1, 2 ... k}$.
I am interested in calculating variance of $$Y = frac {sum_{i=1}^k x_iD_i}{sum_{i=1}^k n_iD_i}$$ where $x_i$ and $n_i$ are given real numbers, $i in {1, 2 ... k}$.
Exactly m of k indicators is equal to 1 and all possible combinations are equally possible.
Also, probability for each combination of choosing these m indicators is $frac {1}{k choose m}$
probability statistics variance
asked Dec 25 '18 at 9:24
dkutlesicdkutlesic
11
$endgroup$
closed as off-topic by Kavi Rama Murthy, Lee David Chung Lin, StubbornAtom, NCh, Namaste Dec 29 '18 at 21:24
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Kavi Rama Murthy, Lee David Chung Lin, StubbornAtom, NCh, Namaste
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Let $D_i$ be indicators, $iin{1, 2 ... k}$.
I am interested in calculating variance of $$Y = frac {sum_{i=1}^k x_iD_i}{sum_{i=1}^k n_iD_i}$$ where $x_i$ and $n_i$ are given real numbers, $i in {1, 2 ... k}$.
Exactly m of k indicators is equal to 1 and all possible combinations are equally possible.
Also, probability for each combination of choosing these m indicators is $frac {1}{k choose m}$
probability statistics variance
asked Dec 25 '18 at 9:24
dkutlesicdkutlesic
11
$endgroup$
Let $D_i$ be indicators, $iin{1, 2 ... k}$.
I am interested in calculating variance of $$Y = frac {sum_{i=1}^k x_iD_i}{sum_{i=1}^k n_iD_i}$$ where $x_i$ and $n_i$ are given real numbers, $i in {1, 2 ... k}$.
Exactly m of k indicators is equal to 1 and all possible combinations are equally possible.
Also, probability for each combination of choosing these m indicators is $frac {1}{k choose m}$
probability statistics variance
probability statistics variance
asked Dec 25 '18 at 9:24
dkutlesicdkutlesic
11
asked Dec 25 '18 at 9:24
dkutlesicdkutlesic
11
edited Dec 25 '18 at 18:01
dkutlesic
asked Dec 25 '18 at 9:24
dkutlesicdkutlesic
11
asked Dec 25 '18 at 9:24
dkutlesicdkutlesic
11
asked Dec 25 '18 at 9:24
dkutlesicdkutlesic
11
11
closed as off-topic by Kavi Rama Murthy, Lee David Chung Lin, StubbornAtom, NCh, Namaste Dec 29 '18 at 21:24
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Kavi Rama Murthy, Lee David Chung Lin, StubbornAtom, NCh, Namaste
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Kavi Rama Murthy, Lee David Chung Lin, StubbornAtom, NCh, Namaste Dec 29 '18 at 21:24
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Kavi Rama Murthy, Lee David Chung Lin, StubbornAtom, NCh, Namaste
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
add a comment |
1 Answer
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$begingroup$
begin{align*}
textrm{Var} left( frac{sum x_i D_i}{sum n_i D_i}right) &= frac{1}{{kchoose m}}sum_{sigma} left{mathbb{E}left[
frac{sum x_i D_i}{sum n_i D_i} right] - frac{sum x_i D_i^{(sigma)}}{sum n_i D_i^{(sigma)}} right}^2 \\
&= frac{1}{{kchoose m}}sum_{sigma} left{frac{1}{{kchoose m}} sum_{sigma'} left[
frac{sum x_i D_i^{(sigma')}}{sum n_i D_i^{(sigma')}} right] - frac{sum x_i D_i^{(sigma)}}{sum n_i D_i^{(sigma)}} right}^2
end{align*}
where the $sigma$ and $sigma'$ sum over all valid combinations of values of indicators.
answered Dec 25 '18 at 18:43
Nikola SamardzicNikola Samardzic
111
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
begin{align*}
textrm{Var} left( frac{sum x_i D_i}{sum n_i D_i}right) &= frac{1}{{kchoose m}}sum_{sigma} left{mathbb{E}left[
frac{sum x_i D_i}{sum n_i D_i} right] - frac{sum x_i D_i^{(sigma)}}{sum n_i D_i^{(sigma)}} right}^2 \\
&= frac{1}{{kchoose m}}sum_{sigma} left{frac{1}{{kchoose m}} sum_{sigma'} left[
frac{sum x_i D_i^{(sigma')}}{sum n_i D_i^{(sigma')}} right] - frac{sum x_i D_i^{(sigma)}}{sum n_i D_i^{(sigma)}} right}^2
end{align*}
where the $sigma$ and $sigma'$ sum over all valid combinations of values of indicators.
answered Dec 25 '18 at 18:43
Nikola SamardzicNikola Samardzic
111
$endgroup$
add a comment |
$begingroup$
begin{align*}
textrm{Var} left( frac{sum x_i D_i}{sum n_i D_i}right) &= frac{1}{{kchoose m}}sum_{sigma} left{mathbb{E}left[
frac{sum x_i D_i}{sum n_i D_i} right] - frac{sum x_i D_i^{(sigma)}}{sum n_i D_i^{(sigma)}} right}^2 \\
&= frac{1}{{kchoose m}}sum_{sigma} left{frac{1}{{kchoose m}} sum_{sigma'} left[
frac{sum x_i D_i^{(sigma')}}{sum n_i D_i^{(sigma')}} right] - frac{sum x_i D_i^{(sigma)}}{sum n_i D_i^{(sigma)}} right}^2
end{align*}
where the $sigma$ and $sigma'$ sum over all valid combinations of values of indicators.
answered Dec 25 '18 at 18:43
Nikola SamardzicNikola Samardzic
111
$endgroup$
add a comment |
$begingroup$
begin{align*}
textrm{Var} left( frac{sum x_i D_i}{sum n_i D_i}right) &= frac{1}{{kchoose m}}sum_{sigma} left{mathbb{E}left[
frac{sum x_i D_i}{sum n_i D_i} right] - frac{sum x_i D_i^{(sigma)}}{sum n_i D_i^{(sigma)}} right}^2 \\
&= frac{1}{{kchoose m}}sum_{sigma} left{frac{1}{{kchoose m}} sum_{sigma'} left[
frac{sum x_i D_i^{(sigma')}}{sum n_i D_i^{(sigma')}} right] - frac{sum x_i D_i^{(sigma)}}{sum n_i D_i^{(sigma)}} right}^2
end{align*}
where the $sigma$ and $sigma'$ sum over all valid combinations of values of indicators.
answered Dec 25 '18 at 18:43
Nikola SamardzicNikola Samardzic
111
$endgroup$
begin{align*}
textrm{Var} left( frac{sum x_i D_i}{sum n_i D_i}right) &= frac{1}{{kchoose m}}sum_{sigma} left{mathbb{E}left[
frac{sum x_i D_i}{sum n_i D_i} right] - frac{sum x_i D_i^{(sigma)}}{sum n_i D_i^{(sigma)}} right}^2 \\
&= frac{1}{{kchoose m}}sum_{sigma} left{frac{1}{{kchoose m}} sum_{sigma'} left[
frac{sum x_i D_i^{(sigma')}}{sum n_i D_i^{(sigma')}} right] - frac{sum x_i D_i^{(sigma)}}{sum n_i D_i^{(sigma)}} right}^2
end{align*}
where the $sigma$ and $sigma'$ sum over all valid combinations of values of indicators.
answered Dec 25 '18 at 18:43
Nikola SamardzicNikola Samardzic
111
answered Dec 25 '18 at 18:43
Nikola SamardzicNikola Samardzic
111
answered Dec 25 '18 at 18:43
Nikola SamardzicNikola Samardzic
111
answered Dec 25 '18 at 18:43
Nikola SamardzicNikola Samardzic
111
111
add a comment |
add a comment |
