The mean value of sample moment with order k
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I've this problem of statistics that I can't resolve. So I hope that someone can halp me.
The problem is this:
Let sampling moments $overline{X_n^k}=frac{1}{n}sum_{i=1}^nX_i^k$, where for $k=1$ $overline{X_n^k}=overline{X_n}$.
Show that $E(overline{X_n^k})=E(X)$, where $E(Y)$ is the mean value of $Y$.
I did this:
$E(overline{X_n^k})=E(frac{1}{n}sum_{i=1}^nX_i^k)=frac{1}{n}E(sum_{i=1}^nX_i^k)$
I think that for ID of $X_i$, I can write:
$frac{1}{n}E(sum_{i=1}^nX_i^k)=frac{1}{n}*nE(X_i^k)=E(X_i^k)$.
Is this correct? And how can I go on? I thought to use the moment generating functions, but I don't know use it.
Thank you for your help.
statistics means sampling
asked Nov 19 at 11:31
user502400
214
add a comment |
up vote
0
down vote
favorite
I've this problem of statistics that I can't resolve. So I hope that someone can halp me.
The problem is this:
Let sampling moments $overline{X_n^k}=frac{1}{n}sum_{i=1}^nX_i^k$, where for $k=1$ $overline{X_n^k}=overline{X_n}$.
Show that $E(overline{X_n^k})=E(X)$, where $E(Y)$ is the mean value of $Y$.
I did this:
$E(overline{X_n^k})=E(frac{1}{n}sum_{i=1}^nX_i^k)=frac{1}{n}E(sum_{i=1}^nX_i^k)$
I think that for ID of $X_i$, I can write:
$frac{1}{n}E(sum_{i=1}^nX_i^k)=frac{1}{n}*nE(X_i^k)=E(X_i^k)$.
Is this correct? And how can I go on? I thought to use the moment generating functions, but I don't know use it.
Thank you for your help.
statistics means sampling
asked Nov 19 at 11:31
user502400
214
Well, $E(overline{X^k})=E(X)$ certainly doesn't hold in general. This, if at all, can only be true for particular value distributions, so we need more information on those.
– Andreas
Nov 19 at 11:35
Did you intend to say "Show that $Eleft(overline{X_n^k}right)=Eleft(X^kright)$" with a $,^k$ on the right hand side?
– Henry
Nov 19 at 17:15
I think that $X_i$ have random variable generative $Xsim N(mu,sigma ^2)$. My teacher didn't specify it, for this reason I didn't write it. But before of this exercise we treated this random variable.
– user502400
Nov 20 at 8:42
I don't know Henry. This is the text that my teacher gave us, but she may have been wrong.
– user502400
Nov 20 at 8:47
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I've this problem of statistics that I can't resolve. So I hope that someone can halp me.
The problem is this:
Let sampling moments $overline{X_n^k}=frac{1}{n}sum_{i=1}^nX_i^k$, where for $k=1$ $overline{X_n^k}=overline{X_n}$.
Show that $E(overline{X_n^k})=E(X)$, where $E(Y)$ is the mean value of $Y$.
I did this:
$E(overline{X_n^k})=E(frac{1}{n}sum_{i=1}^nX_i^k)=frac{1}{n}E(sum_{i=1}^nX_i^k)$
I think that for ID of $X_i$, I can write:
$frac{1}{n}E(sum_{i=1}^nX_i^k)=frac{1}{n}*nE(X_i^k)=E(X_i^k)$.
Is this correct? And how can I go on? I thought to use the moment generating functions, but I don't know use it.
Thank you for your help.
statistics means sampling
asked Nov 19 at 11:31
user502400
214
I've this problem of statistics that I can't resolve. So I hope that someone can halp me.
The problem is this:
Let sampling moments $overline{X_n^k}=frac{1}{n}sum_{i=1}^nX_i^k$, where for $k=1$ $overline{X_n^k}=overline{X_n}$.
Show that $E(overline{X_n^k})=E(X)$, where $E(Y)$ is the mean value of $Y$.
I did this:
$E(overline{X_n^k})=E(frac{1}{n}sum_{i=1}^nX_i^k)=frac{1}{n}E(sum_{i=1}^nX_i^k)$
I think that for ID of $X_i$, I can write:
$frac{1}{n}E(sum_{i=1}^nX_i^k)=frac{1}{n}*nE(X_i^k)=E(X_i^k)$.
Is this correct? And how can I go on? I thought to use the moment generating functions, but I don't know use it.
Thank you for your help.
statistics means sampling
statistics means sampling
asked Nov 19 at 11:31
user502400
214
asked Nov 19 at 11:31
user502400
214
asked Nov 19 at 11:31
user502400
214
asked Nov 19 at 11:31
user502400
214
asked Nov 19 at 11:31
user502400
214
214
Well, $E(overline{X^k})=E(X)$ certainly doesn't hold in general. This, if at all, can only be true for particular value distributions, so we need more information on those.
– Andreas
Nov 19 at 11:35
Did you intend to say "Show that $Eleft(overline{X_n^k}right)=Eleft(X^kright)$" with a $,^k$ on the right hand side?
– Henry
Nov 19 at 17:15
I think that $X_i$ have random variable generative $Xsim N(mu,sigma ^2)$. My teacher didn't specify it, for this reason I didn't write it. But before of this exercise we treated this random variable.
– user502400
Nov 20 at 8:42
I don't know Henry. This is the text that my teacher gave us, but she may have been wrong.
– user502400
Nov 20 at 8:47
add a comment |
Well, $E(overline{X^k})=E(X)$ certainly doesn't hold in general. This, if at all, can only be true for particular value distributions, so we need more information on those.
– Andreas
Nov 19 at 11:35
Did you intend to say "Show that $Eleft(overline{X_n^k}right)=Eleft(X^kright)$" with a $,^k$ on the right hand side?
– Henry
Nov 19 at 17:15
I think that $X_i$ have random variable generative $Xsim N(mu,sigma ^2)$. My teacher didn't specify it, for this reason I didn't write it. But before of this exercise we treated this random variable.
– user502400
Nov 20 at 8:42
I don't know Henry. This is the text that my teacher gave us, but she may have been wrong.
– user502400
Nov 20 at 8:47
Well, $E(overline{X^k})=E(X)$ certainly doesn't hold in general. This, if at all, can only be true for particular value distributions, so we need more information on those.
– Andreas
Nov 19 at 11:35
Well, $E(overline{X^k})=E(X)$ certainly doesn't hold in general. This, if at all, can only be true for particular value distributions, so we need more information on those.
– Andreas
Nov 19 at 11:35
Did you intend to say "Show that $Eleft(overline{X_n^k}right)=Eleft(X^kright)$" with a $,^k$ on the right hand side?
– Henry
Nov 19 at 17:15
Did you intend to say "Show that $Eleft(overline{X_n^k}right)=Eleft(X^kright)$" with a $,^k$ on the right hand side?
– Henry
Nov 19 at 17:15
I think that $X_i$ have random variable generative $Xsim N(mu,sigma ^2)$. My teacher didn't specify it, for this reason I didn't write it. But before of this exercise we treated this random variable.
– user502400
Nov 20 at 8:42
I think that $X_i$ have random variable generative $Xsim N(mu,sigma ^2)$. My teacher didn't specify it, for this reason I didn't write it. But before of this exercise we treated this random variable.
– user502400
Nov 20 at 8:42
I don't know Henry. This is the text that my teacher gave us, but she may have been wrong.
– user502400
Nov 20 at 8:47
I don't know Henry. This is the text that my teacher gave us, but she may have been wrong.
– user502400
Nov 20 at 8:47
add a comment |
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Well, $E(overline{X^k})=E(X)$ certainly doesn't hold in general. This, if at all, can only be true for particular value distributions, so we need more information on those.
– Andreas
Nov 19 at 11:35
Did you intend to say "Show that $Eleft(overline{X_n^k}right)=Eleft(X^kright)$" with a $,^k$ on the right hand side?
– Henry
Nov 19 at 17:15
I think that $X_i$ have random variable generative $Xsim N(mu,sigma ^2)$. My teacher didn't specify it, for this reason I didn't write it. But before of this exercise we treated this random variable.
– user502400
Nov 20 at 8:42
I don't know Henry. This is the text that my teacher gave us, but she may have been wrong.
– user502400
Nov 20 at 8:47