Mixing
Simple correlation question
$begingroup$
Let $Z$ and $Y$ be standard normal and independent random varibales. Define $X := sqrt{p} Z + sqrt{1-p} Y$. What is the correlation between X and Z?
My approach:
Corr(X, Z) = $E(XZ) /1 = E(sqrt{p} Z^2) / 1 = sqrt{p}$
However, a book I am reading says that the correlation is $p$. Am I missing something?
probability statistics
asked Jan 7 at 11:23
master_goonmaster_goon
1588
$endgroup$
add a comment |
$begingroup$
Let $Z$ and $Y$ be standard normal and independent random varibales. Define $X := sqrt{p} Z + sqrt{1-p} Y$. What is the correlation between X and Z?
My approach:
Corr(X, Z) = $E(XZ) /1 = E(sqrt{p} Z^2) / 1 = sqrt{p}$
However, a book I am reading says that the correlation is $p$. Am I missing something?
probability statistics
asked Jan 7 at 11:23
master_goonmaster_goon
1588
$endgroup$
$begingroup$
Your answer is correct by the way.
$endgroup$
– StubbornAtom
Jan 7 at 11:55
$begingroup$
The book is wrong, unless you misread something. Perhaps the linear combination was meant to be $X := sqrt{p^2} Z + sqrt{1-p^2} Y$?
$endgroup$
– leonbloy
Jan 7 at 14:42
$begingroup$
Just to concur with the other comments: The usual way to define simple serial correlation in a time series is $X_i=rho X_{i-1}+sqrt{1-rho^2} Z_i$ so $sqrt{rho}$ is almost certainly a typo and should simply be $rho$.
$endgroup$
– JimB
Jan 7 at 15:12
add a comment |
$begingroup$
Let $Z$ and $Y$ be standard normal and independent random varibales. Define $X := sqrt{p} Z + sqrt{1-p} Y$. What is the correlation between X and Z?
My approach:
Corr(X, Z) = $E(XZ) /1 = E(sqrt{p} Z^2) / 1 = sqrt{p}$
However, a book I am reading says that the correlation is $p$. Am I missing something?
probability statistics
asked Jan 7 at 11:23
master_goonmaster_goon
1588
$endgroup$
Let $Z$ and $Y$ be standard normal and independent random varibales. Define $X := sqrt{p} Z + sqrt{1-p} Y$. What is the correlation between X and Z?
My approach:
Corr(X, Z) = $E(XZ) /1 = E(sqrt{p} Z^2) / 1 = sqrt{p}$
However, a book I am reading says that the correlation is $p$. Am I missing something?
probability statistics
probability statistics
asked Jan 7 at 11:23
master_goonmaster_goon
1588
asked Jan 7 at 11:23
master_goonmaster_goon
1588
asked Jan 7 at 11:23
master_goonmaster_goon
1588
asked Jan 7 at 11:23
master_goonmaster_goon
1588
asked Jan 7 at 11:23
master_goonmaster_goon
1588
1588
$begingroup$
Your answer is correct by the way.
$endgroup$
– StubbornAtom
Jan 7 at 11:55
$begingroup$
The book is wrong, unless you misread something. Perhaps the linear combination was meant to be $X := sqrt{p^2} Z + sqrt{1-p^2} Y$?
$endgroup$
– leonbloy
Jan 7 at 14:42
$begingroup$
Just to concur with the other comments: The usual way to define simple serial correlation in a time series is $X_i=rho X_{i-1}+sqrt{1-rho^2} Z_i$ so $sqrt{rho}$ is almost certainly a typo and should simply be $rho$.
$endgroup$
– JimB
Jan 7 at 15:12
add a comment |
$begingroup$
Your answer is correct by the way.
$endgroup$
– StubbornAtom
Jan 7 at 11:55
$begingroup$
The book is wrong, unless you misread something. Perhaps the linear combination was meant to be $X := sqrt{p^2} Z + sqrt{1-p^2} Y$?
$endgroup$
– leonbloy
Jan 7 at 14:42
$begingroup$
Just to concur with the other comments: The usual way to define simple serial correlation in a time series is $X_i=rho X_{i-1}+sqrt{1-rho^2} Z_i$ so $sqrt{rho}$ is almost certainly a typo and should simply be $rho$.
$endgroup$
– JimB
Jan 7 at 15:12
$begingroup$
Your answer is correct by the way.
$endgroup$
– StubbornAtom
Jan 7 at 11:55
$begingroup$
Your answer is correct by the way.
$endgroup$
– StubbornAtom
Jan 7 at 11:55
$begingroup$
The book is wrong, unless you misread something. Perhaps the linear combination was meant to be $X := sqrt{p^2} Z + sqrt{1-p^2} Y$?
$endgroup$
– leonbloy
Jan 7 at 14:42
$begingroup$
The book is wrong, unless you misread something. Perhaps the linear combination was meant to be $X := sqrt{p^2} Z + sqrt{1-p^2} Y$?
$endgroup$
– leonbloy
Jan 7 at 14:42
$begingroup$
Just to concur with the other comments: The usual way to define simple serial correlation in a time series is $X_i=rho X_{i-1}+sqrt{1-rho^2} Z_i$ so $sqrt{rho}$ is almost certainly a typo and should simply be $rho$.
$endgroup$
– JimB
Jan 7 at 15:12
$begingroup$
Just to concur with the other comments: The usual way to define simple serial correlation in a time series is $X_i=rho X_{i-1}+sqrt{1-rho^2} Z_i$ so $sqrt{rho}$ is almost certainly a typo and should simply be $rho$.
$endgroup$
– JimB
Jan 7 at 15:12
add a comment |
0
active
oldest
votes
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%2f3064895%2fsimple-correlation-question%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
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%2f3064895%2fsimple-correlation-question%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
$begingroup$
Your answer is correct by the way.
$endgroup$
– StubbornAtom
Jan 7 at 11:55
$begingroup$
The book is wrong, unless you misread something. Perhaps the linear combination was meant to be $X := sqrt{p^2} Z + sqrt{1-p^2} Y$?
$endgroup$
– leonbloy
Jan 7 at 14:42
$begingroup$
Just to concur with the other comments: The usual way to define simple serial correlation in a time series is $X_i=rho X_{i-1}+sqrt{1-rho^2} Z_i$ so $sqrt{rho}$ is almost certainly a typo and should simply be $rho$.
$endgroup$
– JimB
Jan 7 at 15:12