$X$ is a random variable, if $Bbb E(X^2)=1$ and $Bbb E(X)geq a>0$, prove that $Bbb P(Xgeqlambda...
$begingroup$
This is a problem in KaiLai Chung's A Course in Probability Theory.
Given a nonnegative random variable $X$ defined on $Omega$, if $mathbb{E}(X^2)=1$ and $mathbb{E}(X)geq a >0$, prove that $$mathbb{P}(Xgeq lambda a)geq (a-lambda a)^2$$
for $0leqlambda leq 1$.
Let $A={xin Omega:X(x)geq lambda a}$, we get
$$int_A (X-lambda a)geq a-int_Alambda a -int_{A^c}X$$
and $$int_A (X^2-lambda^2 a^2)=1-int_Alambda^2a^2-int_{A^c}X^2$$
I want to contrast $int_A (X-lambda a)$ and $int_A (X^2-lambda^2 a^2)$, but I don't know how to do it, could anyone gives me some hints?
probability integration lp-spaces
$endgroup$
add a comment |
$begingroup$
This is a problem in KaiLai Chung's A Course in Probability Theory.
Given a nonnegative random variable $X$ defined on $Omega$, if $mathbb{E}(X^2)=1$ and $mathbb{E}(X)geq a >0$, prove that $$mathbb{P}(Xgeq lambda a)geq (a-lambda a)^2$$
for $0leqlambda leq 1$.
Let $A={xin Omega:X(x)geq lambda a}$, we get
$$int_A (X-lambda a)geq a-int_Alambda a -int_{A^c}X$$
and $$int_A (X^2-lambda^2 a^2)=1-int_Alambda^2a^2-int_{A^c}X^2$$
I want to contrast $int_A (X-lambda a)$ and $int_A (X^2-lambda^2 a^2)$, but I don't know how to do it, could anyone gives me some hints?
probability integration lp-spaces
$endgroup$
$begingroup$
Chebyshev might be useful.
$endgroup$
– copper.hat
Mar 28 at 3:30
add a comment |
$begingroup$
This is a problem in KaiLai Chung's A Course in Probability Theory.
Given a nonnegative random variable $X$ defined on $Omega$, if $mathbb{E}(X^2)=1$ and $mathbb{E}(X)geq a >0$, prove that $$mathbb{P}(Xgeq lambda a)geq (a-lambda a)^2$$
for $0leqlambda leq 1$.
Let $A={xin Omega:X(x)geq lambda a}$, we get
$$int_A (X-lambda a)geq a-int_Alambda a -int_{A^c}X$$
and $$int_A (X^2-lambda^2 a^2)=1-int_Alambda^2a^2-int_{A^c}X^2$$
I want to contrast $int_A (X-lambda a)$ and $int_A (X^2-lambda^2 a^2)$, but I don't know how to do it, could anyone gives me some hints?
probability integration lp-spaces
$endgroup$
This is a problem in KaiLai Chung's A Course in Probability Theory.
Given a nonnegative random variable $X$ defined on $Omega$, if $mathbb{E}(X^2)=1$ and $mathbb{E}(X)geq a >0$, prove that $$mathbb{P}(Xgeq lambda a)geq (a-lambda a)^2$$
for $0leqlambda leq 1$.
Let $A={xin Omega:X(x)geq lambda a}$, we get
$$int_A (X-lambda a)geq a-int_Alambda a -int_{A^c}X$$
and $$int_A (X^2-lambda^2 a^2)=1-int_Alambda^2a^2-int_{A^c}X^2$$
I want to contrast $int_A (X-lambda a)$ and $int_A (X^2-lambda^2 a^2)$, but I don't know how to do it, could anyone gives me some hints?
probability integration lp-spaces
probability integration lp-spaces
edited Mar 28 at 9:19
Asaf Karagila♦
307k33440773
307k33440773
asked Mar 28 at 3:11
Xin FuXin Fu
34319
34319
$begingroup$
Chebyshev might be useful.
$endgroup$
– copper.hat
Mar 28 at 3:30
add a comment |
$begingroup$
Chebyshev might be useful.
$endgroup$
– copper.hat
Mar 28 at 3:30
$begingroup$
Chebyshev might be useful.
$endgroup$
– copper.hat
Mar 28 at 3:30
$begingroup$
Chebyshev might be useful.
$endgroup$
– copper.hat
Mar 28 at 3:30
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You have
$$
alemathbb E(X) = int_{Xlelambda a}X,dP + int_{Xgelambda a}X,dP,le,lambda a + int_{Xgelambda a}X,dP.
$$
Hence,
$$
a(1-lambda),le,int_{Xgelambda a}X,dP,le,left(int_{Xgelambda a}X^2,dPright)^{1/2}cdot P(Xgelambda a)^{1/2},le,P(Xgelambda a)^{1/2}.
$$
Square this and you're done.
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
Mar 28 at 3:44
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%2f3165418%2fx-is-a-random-variable-if-bbb-ex2-1-and-bbb-ex-geq-a0-prove-that%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
$begingroup$
You have
$$
alemathbb E(X) = int_{Xlelambda a}X,dP + int_{Xgelambda a}X,dP,le,lambda a + int_{Xgelambda a}X,dP.
$$
Hence,
$$
a(1-lambda),le,int_{Xgelambda a}X,dP,le,left(int_{Xgelambda a}X^2,dPright)^{1/2}cdot P(Xgelambda a)^{1/2},le,P(Xgelambda a)^{1/2}.
$$
Square this and you're done.
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
Mar 28 at 3:44
add a comment |
$begingroup$
You have
$$
alemathbb E(X) = int_{Xlelambda a}X,dP + int_{Xgelambda a}X,dP,le,lambda a + int_{Xgelambda a}X,dP.
$$
Hence,
$$
a(1-lambda),le,int_{Xgelambda a}X,dP,le,left(int_{Xgelambda a}X^2,dPright)^{1/2}cdot P(Xgelambda a)^{1/2},le,P(Xgelambda a)^{1/2}.
$$
Square this and you're done.
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
Mar 28 at 3:44
add a comment |
$begingroup$
You have
$$
alemathbb E(X) = int_{Xlelambda a}X,dP + int_{Xgelambda a}X,dP,le,lambda a + int_{Xgelambda a}X,dP.
$$
Hence,
$$
a(1-lambda),le,int_{Xgelambda a}X,dP,le,left(int_{Xgelambda a}X^2,dPright)^{1/2}cdot P(Xgelambda a)^{1/2},le,P(Xgelambda a)^{1/2}.
$$
Square this and you're done.
$endgroup$
You have
$$
alemathbb E(X) = int_{Xlelambda a}X,dP + int_{Xgelambda a}X,dP,le,lambda a + int_{Xgelambda a}X,dP.
$$
Hence,
$$
a(1-lambda),le,int_{Xgelambda a}X,dP,le,left(int_{Xgelambda a}X^2,dPright)^{1/2}cdot P(Xgelambda a)^{1/2},le,P(Xgelambda a)^{1/2}.
$$
Square this and you're done.
answered Mar 28 at 3:37
amsmathamsmath
3,273420
3,273420
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
Mar 28 at 3:44
add a comment |
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
Mar 28 at 3:44
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
Mar 28 at 3:44
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
Mar 28 at 3:44
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%2f3165418%2fx-is-a-random-variable-if-bbb-ex2-1-and-bbb-ex-geq-a0-prove-that%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
$begingroup$
Chebyshev might be useful.
$endgroup$
– copper.hat
Mar 28 at 3:30