Mixing
The expected-value of the square of Sample Variance.
Suppose $X_1, cdots, X_n$ are i.d.d. samples from population $X sim N(mu,sigma^2)$, and the sample variance is denoted by
$T = sum_{i = 1}^n frac{(X_i - overline{X})^2}{n}$.
I am curious about the expected-value of $T^2$, which is the square of $T$.
Apparently the key problem is what the distribution of $T^2$ is ?
According to my intuition, it may be some kind of F-distribution, but how to prove it ,especially to solve the cross term is the biggest problem that I have encountered.
probability expected-value
asked Nov 13 '18 at 7:49
Maxius Xu
113
add a comment |
Suppose $X_1, cdots, X_n$ are i.d.d. samples from population $X sim N(mu,sigma^2)$, and the sample variance is denoted by
$T = sum_{i = 1}^n frac{(X_i - overline{X})^2}{n}$.
I am curious about the expected-value of $T^2$, which is the square of $T$.
Apparently the key problem is what the distribution of $T^2$ is ?
According to my intuition, it may be some kind of F-distribution, but how to prove it ,especially to solve the cross term is the biggest problem that I have encountered.
probability expected-value
asked Nov 13 '18 at 7:49
Maxius Xu
113
en.wikipedia.org/wiki/… Both normal case, and the general case are given, for the variance of the sample variance. Then you add back $sigma^4$ to obtain the desired moment. Usually we denote $S^2$ to be the sample variance and $S$ for the sample standard deviation.
– BGM
Nov 13 '18 at 8:17
What I want is the "square" of the sample variance.
– Maxius Xu
Nov 13 '18 at 11:27
add a comment |
Suppose $X_1, cdots, X_n$ are i.d.d. samples from population $X sim N(mu,sigma^2)$, and the sample variance is denoted by
$T = sum_{i = 1}^n frac{(X_i - overline{X})^2}{n}$.
I am curious about the expected-value of $T^2$, which is the square of $T$.
Apparently the key problem is what the distribution of $T^2$ is ?
According to my intuition, it may be some kind of F-distribution, but how to prove it ,especially to solve the cross term is the biggest problem that I have encountered.
probability expected-value
asked Nov 13 '18 at 7:49
Maxius Xu
113
Suppose $X_1, cdots, X_n$ are i.d.d. samples from population $X sim N(mu,sigma^2)$, and the sample variance is denoted by
$T = sum_{i = 1}^n frac{(X_i - overline{X})^2}{n}$.
I am curious about the expected-value of $T^2$, which is the square of $T$.
Apparently the key problem is what the distribution of $T^2$ is ?
According to my intuition, it may be some kind of F-distribution, but how to prove it ,especially to solve the cross term is the biggest problem that I have encountered.
probability expected-value
probability expected-value
asked Nov 13 '18 at 7:49
Maxius Xu
113
asked Nov 13 '18 at 7:49
Maxius Xu
113
edited Nov 13 '18 at 11:26
asked Nov 13 '18 at 7:49
Maxius Xu
113
asked Nov 13 '18 at 7:49
Maxius Xu
113
asked Nov 13 '18 at 7:49
Maxius Xu
113
113
en.wikipedia.org/wiki/… Both normal case, and the general case are given, for the variance of the sample variance. Then you add back $sigma^4$ to obtain the desired moment. Usually we denote $S^2$ to be the sample variance and $S$ for the sample standard deviation.
– BGM
Nov 13 '18 at 8:17
What I want is the "square" of the sample variance.
– Maxius Xu
Nov 13 '18 at 11:27
add a comment |
en.wikipedia.org/wiki/… Both normal case, and the general case are given, for the variance of the sample variance. Then you add back $sigma^4$ to obtain the desired moment. Usually we denote $S^2$ to be the sample variance and $S$ for the sample standard deviation.
– BGM
Nov 13 '18 at 8:17
What I want is the "square" of the sample variance.
– Maxius Xu
Nov 13 '18 at 11:27
en.wikipedia.org/wiki/… Both normal case, and the general case are given, for the variance of the sample variance. Then you add back $sigma^4$ to obtain the desired moment. Usually we denote $S^2$ to be the sample variance and $S$ for the sample standard deviation.
– BGM
Nov 13 '18 at 8:17
en.wikipedia.org/wiki/… Both normal case, and the general case are given, for the variance of the sample variance. Then you add back $sigma^4$ to obtain the desired moment. Usually we denote $S^2$ to be the sample variance and $S$ for the sample standard deviation.
– BGM
Nov 13 '18 at 8:17
What I want is the "square" of the sample variance.
– Maxius Xu
Nov 13 '18 at 11:27
What I want is the "square" of the sample variance.
– Maxius Xu
Nov 13 '18 at 11:27
add a comment |
1 Answer
1
active
oldest
votes
You might now this forumla:
$$
text{Var}[X] = E[X^2] - E[X]^2
$$
I.e.
$$
E[X^2] = text{Var}[X] + E[X]^2
$$
The variance is the expected value of the squared variable, but centered at its expected value.
In this case, the random variable is the sample distribution, which has a Chi-squared distribution – see the link in the comment.
answered Nov 20 '18 at 14:40
Slug Pue
2,15111020
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2996459%2fthe-expected-value-of-the-square-of-sample-variance%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
You might now this forumla:
$$
text{Var}[X] = E[X^2] - E[X]^2
$$
I.e.
$$
E[X^2] = text{Var}[X] + E[X]^2
$$
The variance is the expected value of the squared variable, but centered at its expected value.
In this case, the random variable is the sample distribution, which has a Chi-squared distribution – see the link in the comment.
answered Nov 20 '18 at 14:40
Slug Pue
2,15111020
add a comment |
You might now this forumla:
$$
text{Var}[X] = E[X^2] - E[X]^2
$$
I.e.
$$
E[X^2] = text{Var}[X] + E[X]^2
$$
The variance is the expected value of the squared variable, but centered at its expected value.
In this case, the random variable is the sample distribution, which has a Chi-squared distribution – see the link in the comment.
answered Nov 20 '18 at 14:40
Slug Pue
2,15111020
add a comment |
You might now this forumla:
$$
text{Var}[X] = E[X^2] - E[X]^2
$$
I.e.
$$
E[X^2] = text{Var}[X] + E[X]^2
$$
The variance is the expected value of the squared variable, but centered at its expected value.
In this case, the random variable is the sample distribution, which has a Chi-squared distribution – see the link in the comment.
answered Nov 20 '18 at 14:40
Slug Pue
2,15111020
You might now this forumla:
$$
text{Var}[X] = E[X^2] - E[X]^2
$$
I.e.
$$
E[X^2] = text{Var}[X] + E[X]^2
$$
The variance is the expected value of the squared variable, but centered at its expected value.
In this case, the random variable is the sample distribution, which has a Chi-squared distribution – see the link in the comment.
answered Nov 20 '18 at 14:40
Slug Pue
2,15111020
answered Nov 20 '18 at 14:40
Slug Pue
2,15111020
answered Nov 20 '18 at 14:40
Slug Pue
2,15111020
answered Nov 20 '18 at 14:40
Slug Pue
2,15111020
2,15111020
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2996459%2fthe-expected-value-of-the-square-of-sample-variance%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
en.wikipedia.org/wiki/… Both normal case, and the general case are given, for the variance of the sample variance. Then you add back $sigma^4$ to obtain the desired moment. Usually we denote $S^2$ to be the sample variance and $S$ for the sample standard deviation.
– BGM
Nov 13 '18 at 8:17
What I want is the "square" of the sample variance.
– Maxius Xu
Nov 13 '18 at 11:27