Leibniz rule when range of integration is defined by inequality
I would like to solve this
$$frac{partialint_{S(x)>S(theta)}(S(x)-S(theta))dF(x)}{partialtheta}$$
where $S$ is a single-valued, differentiable, and strictly increasing function and $F$ is a distribution function.
This is more advanced than the Leibniz rule explained in wikipedia or any page that comes up easily. In those pages, range of integration is defined easily as interval.
More generally, I want to solve when there are more than 1 variable, but differentiation is taken by one variable:
$$frac{partialint_{S(x_{1},x_{2})>S(theta_{1},theta_{2})}(S(x_{1},x_{2})-S(theta_{1},theta_{2}))dF(x_{1},x_{2})}{partialtheta_{1}}$$
Does it need to be zero? At least that should be what I will be getting to be consistent with the paper I am reading. I tried to apply the concept in the Leibniz rule as it is defined online but it doesn't seem to be straightforward.
calculus integration derivatives lebesgue-integral contour-integration
asked 2 days ago
user42459user42459
1286
|
show 8 more comments
I would like to solve this
$$frac{partialint_{S(x)>S(theta)}(S(x)-S(theta))dF(x)}{partialtheta}$$
where $S$ is a single-valued, differentiable, and strictly increasing function and $F$ is a distribution function.
This is more advanced than the Leibniz rule explained in wikipedia or any page that comes up easily. In those pages, range of integration is defined easily as interval.
More generally, I want to solve when there are more than 1 variable, but differentiation is taken by one variable:
$$frac{partialint_{S(x_{1},x_{2})>S(theta_{1},theta_{2})}(S(x_{1},x_{2})-S(theta_{1},theta_{2}))dF(x_{1},x_{2})}{partialtheta_{1}}$$
Does it need to be zero? At least that should be what I will be getting to be consistent with the paper I am reading. I tried to apply the concept in the Leibniz rule as it is defined online but it doesn't seem to be straightforward.
calculus integration derivatives lebesgue-integral contour-integration
asked 2 days ago
user42459user42459
1286
How are $S$ and $F$ defined?
– Matt Samuel
2 days ago
@MattSamuel S is a function. F is a distribution function. So its derivative will bef
– user42459
2 days ago
I can choose $S$ that will make the formula nonsense because the derivative will not exist. So you need to be more specific about the function.
– Matt Samuel
2 days ago
@MattSamuel Thank you. It is single-valued, differentiable, and strictly increasing.
– user42459
2 days ago
Then the integral is over the interval $[theta, infty) $, isn't it?
– Matt Samuel
2 days ago
|
show 8 more comments
I would like to solve this
$$frac{partialint_{S(x)>S(theta)}(S(x)-S(theta))dF(x)}{partialtheta}$$
where $S$ is a single-valued, differentiable, and strictly increasing function and $F$ is a distribution function.
This is more advanced than the Leibniz rule explained in wikipedia or any page that comes up easily. In those pages, range of integration is defined easily as interval.
More generally, I want to solve when there are more than 1 variable, but differentiation is taken by one variable:
$$frac{partialint_{S(x_{1},x_{2})>S(theta_{1},theta_{2})}(S(x_{1},x_{2})-S(theta_{1},theta_{2}))dF(x_{1},x_{2})}{partialtheta_{1}}$$
Does it need to be zero? At least that should be what I will be getting to be consistent with the paper I am reading. I tried to apply the concept in the Leibniz rule as it is defined online but it doesn't seem to be straightforward.
calculus integration derivatives lebesgue-integral contour-integration
asked 2 days ago
user42459user42459
1286
I would like to solve this
$$frac{partialint_{S(x)>S(theta)}(S(x)-S(theta))dF(x)}{partialtheta}$$
where $S$ is a single-valued, differentiable, and strictly increasing function and $F$ is a distribution function.
This is more advanced than the Leibniz rule explained in wikipedia or any page that comes up easily. In those pages, range of integration is defined easily as interval.
More generally, I want to solve when there are more than 1 variable, but differentiation is taken by one variable:
$$frac{partialint_{S(x_{1},x_{2})>S(theta_{1},theta_{2})}(S(x_{1},x_{2})-S(theta_{1},theta_{2}))dF(x_{1},x_{2})}{partialtheta_{1}}$$
Does it need to be zero? At least that should be what I will be getting to be consistent with the paper I am reading. I tried to apply the concept in the Leibniz rule as it is defined online but it doesn't seem to be straightforward.
calculus integration derivatives lebesgue-integral contour-integration
calculus integration derivatives lebesgue-integral contour-integration
asked 2 days ago
user42459user42459
1286
asked 2 days ago
user42459user42459
1286
edited 2 days ago
user42459
asked 2 days ago
user42459user42459
1286
asked 2 days ago
user42459user42459
1286
asked 2 days ago
user42459user42459
1286
1286
How are $S$ and $F$ defined?
– Matt Samuel
2 days ago
@MattSamuel S is a function. F is a distribution function. So its derivative will bef
– user42459
2 days ago
I can choose $S$ that will make the formula nonsense because the derivative will not exist. So you need to be more specific about the function.
– Matt Samuel
2 days ago
@MattSamuel Thank you. It is single-valued, differentiable, and strictly increasing.
– user42459
2 days ago
Then the integral is over the interval $[theta, infty) $, isn't it?
– Matt Samuel
2 days ago
|
show 8 more comments
How are $S$ and $F$ defined?
– Matt Samuel
2 days ago
@MattSamuel S is a function. F is a distribution function. So its derivative will bef
– user42459
2 days ago
I can choose $S$ that will make the formula nonsense because the derivative will not exist. So you need to be more specific about the function.
– Matt Samuel
2 days ago
@MattSamuel Thank you. It is single-valued, differentiable, and strictly increasing.
– user42459
2 days ago
Then the integral is over the interval $[theta, infty) $, isn't it?
– Matt Samuel
2 days ago
How are $S$ and $F$ defined?
– Matt Samuel
2 days ago
How are $S$ and $F$ defined?
– Matt Samuel
2 days ago
@MattSamuel S is a function. F is a distribution function. So its derivative will be
f– user42459
2 days ago
@MattSamuel S is a function. F is a distribution function. So its derivative will be
f– user42459
2 days ago
I can choose $S$ that will make the formula nonsense because the derivative will not exist. So you need to be more specific about the function.
– Matt Samuel
2 days ago
I can choose $S$ that will make the formula nonsense because the derivative will not exist. So you need to be more specific about the function.
– Matt Samuel
2 days ago
@MattSamuel Thank you. It is single-valued, differentiable, and strictly increasing.
– user42459
2 days ago
@MattSamuel Thank you. It is single-valued, differentiable, and strictly increasing.
– user42459
2 days ago
Then the integral is over the interval $[theta, infty) $, isn't it?
– Matt Samuel
2 days ago
Then the integral is over the interval $[theta, infty) $, isn't it?
– Matt Samuel
2 days ago
|
show 8 more comments
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "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%2f3063180%2fleibniz-rule-when-range-of-integration-is-defined-by-inequality%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.
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%2f3063180%2fleibniz-rule-when-range-of-integration-is-defined-by-inequality%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
How are $S$ and $F$ defined?
– Matt Samuel
2 days ago
@MattSamuel S is a function. F is a distribution function. So its derivative will be
f– user42459
2 days ago
I can choose $S$ that will make the formula nonsense because the derivative will not exist. So you need to be more specific about the function.
– Matt Samuel
2 days ago
@MattSamuel Thank you. It is single-valued, differentiable, and strictly increasing.
– user42459
2 days ago
Then the integral is over the interval $[theta, infty) $, isn't it?
– Matt Samuel
2 days ago