How to show that $sum_{i=1}^m (X_i−X_m)^2$ and $sum_{i=1}^n(Y_i− Y_n)^2$ are independent
Let $X_1,...,X_m$ be i.i.d. sample with $N(mu_1,sigma^2)$, and $Y_1,...,Y_n$ be i.i.d. sample with $N(mu_2,2sigma^2)$.
Let $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$.
(b) Determine the values of $alpha$ and $beta$ for which $alpha S_x^2 + beta S_y^2$ will be an unbiased estimator with minimum variance.
It is not so difficult to solve if $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$ are independent. However I don't know how to show that they are independent... any possible helps??
independence sampling
edited Nov 27 at 15:35
amWhy
191k28224439
asked Nov 27 at 14:17
Newt
207
add a comment |
Let $X_1,...,X_m$ be i.i.d. sample with $N(mu_1,sigma^2)$, and $Y_1,...,Y_n$ be i.i.d. sample with $N(mu_2,2sigma^2)$.
Let $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$.
(b) Determine the values of $alpha$ and $beta$ for which $alpha S_x^2 + beta S_y^2$ will be an unbiased estimator with minimum variance.
It is not so difficult to solve if $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$ are independent. However I don't know how to show that they are independent... any possible helps??
independence sampling
edited Nov 27 at 15:35
amWhy
191k28224439
asked Nov 27 at 14:17
Newt
207
Are ${X_1,dots,X_m,Y_1,dots,Y_n}$ independent? If they are, then it's trivial.
– Federico
Nov 27 at 14:33
Xi are independent and Yi are independent. But nothing's known about the independence between Xi and Yi
– Newt
Nov 27 at 14:37
I think its pretty obvious from the statement that $X_i, Y_j$ are assumed to be independent.
– leonbloy
Nov 27 at 15:39
add a comment |
Let $X_1,...,X_m$ be i.i.d. sample with $N(mu_1,sigma^2)$, and $Y_1,...,Y_n$ be i.i.d. sample with $N(mu_2,2sigma^2)$.
Let $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$.
(b) Determine the values of $alpha$ and $beta$ for which $alpha S_x^2 + beta S_y^2$ will be an unbiased estimator with minimum variance.
It is not so difficult to solve if $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$ are independent. However I don't know how to show that they are independent... any possible helps??
independence sampling
edited Nov 27 at 15:35
amWhy
191k28224439
asked Nov 27 at 14:17
Newt
207
Let $X_1,...,X_m$ be i.i.d. sample with $N(mu_1,sigma^2)$, and $Y_1,...,Y_n$ be i.i.d. sample with $N(mu_2,2sigma^2)$.
Let $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$.
(b) Determine the values of $alpha$ and $beta$ for which $alpha S_x^2 + beta S_y^2$ will be an unbiased estimator with minimum variance.
It is not so difficult to solve if $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$ are independent. However I don't know how to show that they are independent... any possible helps??
independence sampling
independence sampling
edited Nov 27 at 15:35
amWhy
191k28224439
asked Nov 27 at 14:17
Newt
207
edited Nov 27 at 15:35
amWhy
191k28224439
asked Nov 27 at 14:17
Newt
207
edited Nov 27 at 15:35
amWhy
191k28224439
edited Nov 27 at 15:35
amWhy
191k28224439
edited Nov 27 at 15:35
amWhy
191k28224439
191k28224439
asked Nov 27 at 14:17
Newt
207
asked Nov 27 at 14:17
Newt
207
asked Nov 27 at 14:17
Newt
207
207
Are ${X_1,dots,X_m,Y_1,dots,Y_n}$ independent? If they are, then it's trivial.
– Federico
Nov 27 at 14:33
Xi are independent and Yi are independent. But nothing's known about the independence between Xi and Yi
– Newt
Nov 27 at 14:37
I think its pretty obvious from the statement that $X_i, Y_j$ are assumed to be independent.
– leonbloy
Nov 27 at 15:39
add a comment |
Are ${X_1,dots,X_m,Y_1,dots,Y_n}$ independent? If they are, then it's trivial.
– Federico
Nov 27 at 14:33
Xi are independent and Yi are independent. But nothing's known about the independence between Xi and Yi
– Newt
Nov 27 at 14:37
I think its pretty obvious from the statement that $X_i, Y_j$ are assumed to be independent.
– leonbloy
Nov 27 at 15:39
Are ${X_1,dots,X_m,Y_1,dots,Y_n}$ independent? If they are, then it's trivial.
– Federico
Nov 27 at 14:33
Are ${X_1,dots,X_m,Y_1,dots,Y_n}$ independent? If they are, then it's trivial.
– Federico
Nov 27 at 14:33
Xi are independent and Yi are independent. But nothing's known about the independence between Xi and Yi
– Newt
Nov 27 at 14:37
Xi are independent and Yi are independent. But nothing's known about the independence between Xi and Yi
– Newt
Nov 27 at 14:37
I think its pretty obvious from the statement that $X_i, Y_j$ are assumed to be independent.
– leonbloy
Nov 27 at 15:39
I think its pretty obvious from the statement that $X_i, Y_j$ are assumed to be independent.
– leonbloy
Nov 27 at 15:39
add a comment |
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Are ${X_1,dots,X_m,Y_1,dots,Y_n}$ independent? If they are, then it's trivial.
– Federico
Nov 27 at 14:33
Xi are independent and Yi are independent. But nothing's known about the independence between Xi and Yi
– Newt
Nov 27 at 14:37
I think its pretty obvious from the statement that $X_i, Y_j$ are assumed to be independent.
– leonbloy
Nov 27 at 15:39