Expected value of inverse of surrounding density
$begingroup$
Fix a probability distribution on a compact set $mathcal{X} subset mathbb{R}$. I wonder what the conditions would be such that
$mathbb{E} left[ frac{1}{F(x+d)-F(x-d)} right]$ does not diverge with $F(cdot)$ being the CDF and/or what would be an example where it diverges. Generally, $d>0$ but in the limit would go to $0$.
We can rewrite as follows
$$
mathbb{E} left[ frac{1}{F(x+d)-F(x-d)} right] = int_mathcal{X} frac{p(x)}{int_{x-d}^{x+d}p(y)dy} dx
$$
I tried numerical integration rules, Hölder's inequality, etc. but have not found a satisfying simple condition. I also could not find a PDF serving as counter-example, yet.
integration probability-theory probability-distributions lebesgue-measure approximate-integration
asked Dec 17 '18 at 10:09
AlexelaAlexela
12
$endgroup$
add a comment |
$begingroup$
Fix a probability distribution on a compact set $mathcal{X} subset mathbb{R}$. I wonder what the conditions would be such that
$mathbb{E} left[ frac{1}{F(x+d)-F(x-d)} right]$ does not diverge with $F(cdot)$ being the CDF and/or what would be an example where it diverges. Generally, $d>0$ but in the limit would go to $0$.
We can rewrite as follows
$$
mathbb{E} left[ frac{1}{F(x+d)-F(x-d)} right] = int_mathcal{X} frac{p(x)}{int_{x-d}^{x+d}p(y)dy} dx
$$
I tried numerical integration rules, Hölder's inequality, etc. but have not found a satisfying simple condition. I also could not find a PDF serving as counter-example, yet.
integration probability-theory probability-distributions lebesgue-measure approximate-integration
asked Dec 17 '18 at 10:09
AlexelaAlexela
12
$endgroup$
add a comment |
$begingroup$
Fix a probability distribution on a compact set $mathcal{X} subset mathbb{R}$. I wonder what the conditions would be such that
$mathbb{E} left[ frac{1}{F(x+d)-F(x-d)} right]$ does not diverge with $F(cdot)$ being the CDF and/or what would be an example where it diverges. Generally, $d>0$ but in the limit would go to $0$.
We can rewrite as follows
$$
mathbb{E} left[ frac{1}{F(x+d)-F(x-d)} right] = int_mathcal{X} frac{p(x)}{int_{x-d}^{x+d}p(y)dy} dx
$$
I tried numerical integration rules, Hölder's inequality, etc. but have not found a satisfying simple condition. I also could not find a PDF serving as counter-example, yet.
integration probability-theory probability-distributions lebesgue-measure approximate-integration
asked Dec 17 '18 at 10:09
AlexelaAlexela
12
$endgroup$
Fix a probability distribution on a compact set $mathcal{X} subset mathbb{R}$. I wonder what the conditions would be such that
$mathbb{E} left[ frac{1}{F(x+d)-F(x-d)} right]$ does not diverge with $F(cdot)$ being the CDF and/or what would be an example where it diverges. Generally, $d>0$ but in the limit would go to $0$.
We can rewrite as follows
$$
mathbb{E} left[ frac{1}{F(x+d)-F(x-d)} right] = int_mathcal{X} frac{p(x)}{int_{x-d}^{x+d}p(y)dy} dx
$$
I tried numerical integration rules, Hölder's inequality, etc. but have not found a satisfying simple condition. I also could not find a PDF serving as counter-example, yet.
integration probability-theory probability-distributions lebesgue-measure approximate-integration
integration probability-theory probability-distributions lebesgue-measure approximate-integration
asked Dec 17 '18 at 10:09
AlexelaAlexela
12
asked Dec 17 '18 at 10:09
AlexelaAlexela
12
asked Dec 17 '18 at 10:09
AlexelaAlexela
12
asked Dec 17 '18 at 10:09
AlexelaAlexela
12
asked Dec 17 '18 at 10:09
AlexelaAlexela
12
12
add a comment |
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%2f3043762%2fexpected-value-of-inverse-of-surrounding-density%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%2f3043762%2fexpected-value-of-inverse-of-surrounding-density%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
