Prove E[x] is the integral of T(x) from 0 to x, where T(x) is the tail distribution function of x
$begingroup$
x is a continuous and nonnegative random variable with cdf F and density f ,prove expectation E[x] is the integral of T(x) from 0 to x, where T(x) is the tail distribution function of x
integration
asked Jan 16 at 1:35
Ben186998Ben186998
32
$endgroup$
add a comment |
$begingroup$
x is a continuous and nonnegative random variable with cdf F and density f ,prove expectation E[x] is the integral of T(x) from 0 to x, where T(x) is the tail distribution function of x
integration
asked Jan 16 at 1:35
Ben186998Ben186998
32
$endgroup$
add a comment |
$begingroup$
x is a continuous and nonnegative random variable with cdf F and density f ,prove expectation E[x] is the integral of T(x) from 0 to x, where T(x) is the tail distribution function of x
integration
asked Jan 16 at 1:35
Ben186998Ben186998
32
$endgroup$
x is a continuous and nonnegative random variable with cdf F and density f ,prove expectation E[x] is the integral of T(x) from 0 to x, where T(x) is the tail distribution function of x
integration
integration
asked Jan 16 at 1:35
Ben186998Ben186998
32
asked Jan 16 at 1:35
Ben186998Ben186998
32
asked Jan 16 at 1:35
Ben186998Ben186998
32
asked Jan 16 at 1:35
Ben186998Ben186998
32
asked Jan 16 at 1:35
Ben186998Ben186998
32
32
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Suppose $X$ is defined on $(Omega, mathcal F,P)$. By Fubini's Theorem we get $int_0^{infty} P{ X>t}, dt =int_0^{infty} int I_{{ X>t}}, dP , dt=int_0^{infty} int I_{{ t<X}}, dt , dP= int X dP =EX$.
answered Jan 16 at 6:06
Kavi Rama MurthyKavi Rama Murthy
71.6k53170
$endgroup$
add a comment |
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%2f3075205%2fprove-ex-is-the-integral-of-tx-from-0-to-x-where-tx-is-the-tail-distribut%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
$begingroup$
Suppose $X$ is defined on $(Omega, mathcal F,P)$. By Fubini's Theorem we get $int_0^{infty} P{ X>t}, dt =int_0^{infty} int I_{{ X>t}}, dP , dt=int_0^{infty} int I_{{ t<X}}, dt , dP= int X dP =EX$.
answered Jan 16 at 6:06
Kavi Rama MurthyKavi Rama Murthy
71.6k53170
$endgroup$
add a comment |
$begingroup$
Suppose $X$ is defined on $(Omega, mathcal F,P)$. By Fubini's Theorem we get $int_0^{infty} P{ X>t}, dt =int_0^{infty} int I_{{ X>t}}, dP , dt=int_0^{infty} int I_{{ t<X}}, dt , dP= int X dP =EX$.
answered Jan 16 at 6:06
Kavi Rama MurthyKavi Rama Murthy
71.6k53170
$endgroup$
add a comment |
$begingroup$
Suppose $X$ is defined on $(Omega, mathcal F,P)$. By Fubini's Theorem we get $int_0^{infty} P{ X>t}, dt =int_0^{infty} int I_{{ X>t}}, dP , dt=int_0^{infty} int I_{{ t<X}}, dt , dP= int X dP =EX$.
answered Jan 16 at 6:06
Kavi Rama MurthyKavi Rama Murthy
71.6k53170
$endgroup$
Suppose $X$ is defined on $(Omega, mathcal F,P)$. By Fubini's Theorem we get $int_0^{infty} P{ X>t}, dt =int_0^{infty} int I_{{ X>t}}, dP , dt=int_0^{infty} int I_{{ t<X}}, dt , dP= int X dP =EX$.
answered Jan 16 at 6:06
Kavi Rama MurthyKavi Rama Murthy
71.6k53170
answered Jan 16 at 6:06
Kavi Rama MurthyKavi Rama Murthy
71.6k53170
answered Jan 16 at 6:06
Kavi Rama MurthyKavi Rama Murthy
71.6k53170
answered Jan 16 at 6:06
Kavi Rama MurthyKavi Rama Murthy
71.6k53170
71.6k53170
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.
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%2f3075205%2fprove-ex-is-the-integral-of-tx-from-0-to-x-where-tx-is-the-tail-distribut%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
