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Variance of $y_i$ if $sum_i=1^10 y_i = 100$ ? Also find $operatornamecov(y_i, y_j)$.
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Let $Y=(Y_1,Y_2,dots,Y_10)$, be a random vector with integer coordinates, uniformly distributed over the set $$left(y_1,dots,y_n);middle.$$ What are $operatornamevar(Y_i)$ and $operatornamecov(Y_i, Y_j), i neq j$?
I tried to solve this problem by finding the distribution of each $y_i$s.
In fact, $f_Y_i(y) = frac108-y choose 8109 choose 9 $.
It is clear that $E(Y_i) = 10 forall i$ as the distributions are identical, so $ sum_i=1^10 E(Y_i) = 100 Rightarrow E(Y_i) = 10 forall i$.
But how do i evaluate $$sum_y=0^100 y^2 frac108-y choose 8109 choose 9$$ to find variance of any $Y_i$?
I also found that if the variance is known, covariance between $Y_i$ and $Y_j$ $(ineq j)$ can be computed from the equation $operatornamevar(sum_i=0^10 Y_i) = 0$
probability probability-distributions
edited Jul 24 at 1:44
Math1000
18.5k31544
asked Jul 19 at 19:54
Pritam
246
 |Â
show 1 more comment
up vote
2
down vote
favorite
Let $Y=(Y_1,Y_2,dots,Y_10)$, be a random vector with integer coordinates, uniformly distributed over the set $$left(y_1,dots,y_n);middle.$$ What are $operatornamevar(Y_i)$ and $operatornamecov(Y_i, Y_j), i neq j$?
I tried to solve this problem by finding the distribution of each $y_i$s.
In fact, $f_Y_i(y) = frac108-y choose 8109 choose 9 $.
It is clear that $E(Y_i) = 10 forall i$ as the distributions are identical, so $ sum_i=1^10 E(Y_i) = 100 Rightarrow E(Y_i) = 10 forall i$.
But how do i evaluate $$sum_y=0^100 y^2 frac108-y choose 8109 choose 9$$ to find variance of any $Y_i$?
I also found that if the variance is known, covariance between $Y_i$ and $Y_j$ $(ineq j)$ can be computed from the equation $operatornamevar(sum_i=0^10 Y_i) = 0$
probability probability-distributions
edited Jul 24 at 1:44
Math1000
18.5k31544
asked Jul 19 at 19:54
Pritam
246
What are the $y_i$'s? Observations? Random variables?
– asdf
Jul 19 at 19:58
$y_i$s are non-negative integers that satisfy the equation
– Pritam
Jul 19 at 20:00
I think i could not express the problem well. $y_i$s are the integer non-negative random variables that will satisfy the equation. Definitely $y_i =10$ is not the only solution. And I want to find the variance between all the values $y_i$ can take. I think I should mention that each solution of this equation is equally likely. Does it get clear then?
– Pritam
Jul 19 at 20:22
Where you have $f_y_i(y),$ you need $f_Y_i(y).$ Inattentiveness to the distinction between $Y_i$ and $y_i$ can lead to confusion or to inability to do some things, starting with making it impossible to understand $Pr(Y_i=y_i). qquad$
– Michael Hardy
Jul 19 at 21:27
$$ beginalign 0 & = operatornamevar(100) = operatornamevar(Y_1 + cdots + Y_10) = sum_i operatornamevar(Y_i) + 2sum_i,j,:, i,<,j operatornamecov (Y_i,Y_j) \ \ & = 10v(Y_1) + 2cdot45operatornamecov(Y_1,Y_2), \ \ endalign $$
– Michael Hardy
Jul 19 at 23:34
 |Â
show 1 more comment
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $Y=(Y_1,Y_2,dots,Y_10)$, be a random vector with integer coordinates, uniformly distributed over the set $$left(y_1,dots,y_n);middle.$$ What are $operatornamevar(Y_i)$ and $operatornamecov(Y_i, Y_j), i neq j$?
I tried to solve this problem by finding the distribution of each $y_i$s.
In fact, $f_Y_i(y) = frac108-y choose 8109 choose 9 $.
It is clear that $E(Y_i) = 10 forall i$ as the distributions are identical, so $ sum_i=1^10 E(Y_i) = 100 Rightarrow E(Y_i) = 10 forall i$.
But how do i evaluate $$sum_y=0^100 y^2 frac108-y choose 8109 choose 9$$ to find variance of any $Y_i$?
I also found that if the variance is known, covariance between $Y_i$ and $Y_j$ $(ineq j)$ can be computed from the equation $operatornamevar(sum_i=0^10 Y_i) = 0$
probability probability-distributions
edited Jul 24 at 1:44
Math1000
18.5k31544
asked Jul 19 at 19:54
Pritam
246
Let $Y=(Y_1,Y_2,dots,Y_10)$, be a random vector with integer coordinates, uniformly distributed over the set $$left(y_1,dots,y_n);middle.$$ What are $operatornamevar(Y_i)$ and $operatornamecov(Y_i, Y_j), i neq j$?
I tried to solve this problem by finding the distribution of each $y_i$s.
In fact, $f_Y_i(y) = frac108-y choose 8109 choose 9 $.
It is clear that $E(Y_i) = 10 forall i$ as the distributions are identical, so $ sum_i=1^10 E(Y_i) = 100 Rightarrow E(Y_i) = 10 forall i$.
But how do i evaluate $$sum_y=0^100 y^2 frac108-y choose 8109 choose 9$$ to find variance of any $Y_i$?
I also found that if the variance is known, covariance between $Y_i$ and $Y_j$ $(ineq j)$ can be computed from the equation $operatornamevar(sum_i=0^10 Y_i) = 0$
probability probability-distributions
edited Jul 24 at 1:44
Math1000
18.5k31544
asked Jul 19 at 19:54
Pritam
246
edited Jul 24 at 1:44
Math1000
18.5k31544
edited Jul 24 at 1:44
Math1000
18.5k31544
edited Jul 24 at 1:44
Math1000
18.5k31544
18.5k31544
asked Jul 19 at 19:54
Pritam
246
asked Jul 19 at 19:54
Pritam
246
asked Jul 19 at 19:54
Pritam
246
246
What are the $y_i$'s? Observations? Random variables?
– asdf
Jul 19 at 19:58
$y_i$s are non-negative integers that satisfy the equation
– Pritam
Jul 19 at 20:00
I think i could not express the problem well. $y_i$s are the integer non-negative random variables that will satisfy the equation. Definitely $y_i =10$ is not the only solution. And I want to find the variance between all the values $y_i$ can take. I think I should mention that each solution of this equation is equally likely. Does it get clear then?
– Pritam
Jul 19 at 20:22
Where you have $f_y_i(y),$ you need $f_Y_i(y).$ Inattentiveness to the distinction between $Y_i$ and $y_i$ can lead to confusion or to inability to do some things, starting with making it impossible to understand $Pr(Y_i=y_i). qquad$
– Michael Hardy
Jul 19 at 21:27
$$ beginalign 0 & = operatornamevar(100) = operatornamevar(Y_1 + cdots + Y_10) = sum_i operatornamevar(Y_i) + 2sum_i,j,:, i,<,j operatornamecov (Y_i,Y_j) \ \ & = 10v(Y_1) + 2cdot45operatornamecov(Y_1,Y_2), \ \ endalign $$
– Michael Hardy
Jul 19 at 23:34
 |Â
show 1 more comment
What are the $y_i$'s? Observations? Random variables?
– asdf
Jul 19 at 19:58
$y_i$s are non-negative integers that satisfy the equation
– Pritam
Jul 19 at 20:00
I think i could not express the problem well. $y_i$s are the integer non-negative random variables that will satisfy the equation. Definitely $y_i =10$ is not the only solution. And I want to find the variance between all the values $y_i$ can take. I think I should mention that each solution of this equation is equally likely. Does it get clear then?
– Pritam
Jul 19 at 20:22
Where you have $f_y_i(y),$ you need $f_Y_i(y).$ Inattentiveness to the distinction between $Y_i$ and $y_i$ can lead to confusion or to inability to do some things, starting with making it impossible to understand $Pr(Y_i=y_i). qquad$
– Michael Hardy
Jul 19 at 21:27
$$ beginalign 0 & = operatornamevar(100) = operatornamevar(Y_1 + cdots + Y_10) = sum_i operatornamevar(Y_i) + 2sum_i,j,:, i,<,j operatornamecov (Y_i,Y_j) \ \ & = 10v(Y_1) + 2cdot45operatornamecov(Y_1,Y_2), \ \ endalign $$
– Michael Hardy
Jul 19 at 23:34
What are the $y_i$'s? Observations? Random variables?
– asdf
Jul 19 at 19:58
What are the $y_i$'s? Observations? Random variables?
– asdf
Jul 19 at 19:58
$y_i$s are non-negative integers that satisfy the equation
– Pritam
Jul 19 at 20:00
$y_i$s are non-negative integers that satisfy the equation
– Pritam
Jul 19 at 20:00
I think i could not express the problem well. $y_i$s are the integer non-negative random variables that will satisfy the equation. Definitely $y_i =10$ is not the only solution. And I want to find the variance between all the values $y_i$ can take. I think I should mention that each solution of this equation is equally likely. Does it get clear then?
– Pritam
Jul 19 at 20:22
I think i could not express the problem well. $y_i$s are the integer non-negative random variables that will satisfy the equation. Definitely $y_i =10$ is not the only solution. And I want to find the variance between all the values $y_i$ can take. I think I should mention that each solution of this equation is equally likely. Does it get clear then?
– Pritam
Jul 19 at 20:22
Where you have $f_y_i(y),$ you need $f_Y_i(y).$ Inattentiveness to the distinction between $Y_i$ and $y_i$ can lead to confusion or to inability to do some things, starting with making it impossible to understand $Pr(Y_i=y_i). qquad$
– Michael Hardy
Jul 19 at 21:27
Where you have $f_y_i(y),$ you need $f_Y_i(y).$ Inattentiveness to the distinction between $Y_i$ and $y_i$ can lead to confusion or to inability to do some things, starting with making it impossible to understand $Pr(Y_i=y_i). qquad$
– Michael Hardy
Jul 19 at 21:27
$$ beginalign 0 & = operatornamevar(100) = operatornamevar(Y_1 + cdots + Y_10) = sum_i operatornamevar(Y_i) + 2sum_i,j,:, i,<,j operatornamecov (Y_i,Y_j) \ \ & = 10v(Y_1) + 2cdot45operatornamecov(Y_1,Y_2), \ \ endalign $$
– Michael Hardy
Jul 19 at 23:34
$$ beginalign 0 & = operatornamevar(100) = operatornamevar(Y_1 + cdots + Y_10) = sum_i operatornamevar(Y_i) + 2sum_i,j,:, i,<,j operatornamecov (Y_i,Y_j) \ \ & = 10v(Y_1) + 2cdot45operatornamecov(Y_1,Y_2), \ \ endalign $$
– Michael Hardy
Jul 19 at 23:34
 |Â
show 1 more comment
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What are the $y_i$'s? Observations? Random variables?
– asdf
Jul 19 at 19:58
$y_i$s are non-negative integers that satisfy the equation
– Pritam
Jul 19 at 20:00
I think i could not express the problem well. $y_i$s are the integer non-negative random variables that will satisfy the equation. Definitely $y_i =10$ is not the only solution. And I want to find the variance between all the values $y_i$ can take. I think I should mention that each solution of this equation is equally likely. Does it get clear then?
– Pritam
Jul 19 at 20:22
Where you have $f_y_i(y),$ you need $f_Y_i(y).$ Inattentiveness to the distinction between $Y_i$ and $y_i$ can lead to confusion or to inability to do some things, starting with making it impossible to understand $Pr(Y_i=y_i). qquad$
– Michael Hardy
Jul 19 at 21:27
$$ beginalign 0 & = operatornamevar(100) = operatornamevar(Y_1 + cdots + Y_10) = sum_i operatornamevar(Y_i) + 2sum_i,j,:, i,<,j operatornamecov (Y_i,Y_j) \ \ & = 10v(Y_1) + 2cdot45operatornamecov(Y_1,Y_2), \ \ endalign $$
– Michael Hardy
Jul 19 at 23:34