A problem in Probability theoryIf $G(x)=P[Xgeq x]$ then $Xgeq c$ is equivalent to $G(X)leq G(c)$ $P$-almost...
How does the UK government determine the size of a mandate?
Would this custom Sorcerer variant that can only learn any verbal-component-only spell be unbalanced?
Is a stroke of luck acceptable after a series of unfavorable events?
How do I extract a value from a time formatted value in excel?
Pre-amplifier input protection
Why not increase contact surface when reentering the atmosphere?
Method to test if a number is a perfect power?
Why didn't Theresa May consult with Parliament before negotiating a deal with the EU?
How does Loki do this?
Is exact Kanji stroke length important?
Was Spock the First Vulcan in Starfleet?
Large drywall patch supports
Purchasing a ticket for someone else in another country?
Where does the Z80 processor start executing from?
Class Action - which options I have?
Was a professor correct to chastise me for writing "Prof. X" rather than "Professor X"?
India just shot down a satellite from the ground. At what altitude range is the resulting debris field?
Sequence of Tenses: Translating the subjunctive
How do we know the LHC results are robust?
Trouble understanding the speech of overseas colleagues
How to write papers efficiently when English isn't my first language?
Lay out the Carpet
What does "I’d sit this one out, Cap," imply or mean in the context?
I'm in charge of equipment buying but no one's ever happy with what I choose. How to fix this?
A problem in Probability theory
If $G(x)=P[Xgeq x]$ then $Xgeq c$ is equivalent to $G(X)leq G(c)$ $P$-almost surelyTrying to establish an inequality on probabilityCan some probability triple give rise to any probability distribution?Expectation of $mathbb{E}(X^{k+1})$Is PDF unique for a random variable $X$ in given probability space?Conditional expectation on different probability measureAverage of Random variables converges in probability.Range of a random variable is measurableIn probability theory what does the notation $int_{Omega} X(omega) P(domega)$ mean?Probability theory: Convergence
$begingroup$
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
asked 3 hours ago
Xin FuXin Fu
1568
$endgroup$
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
asked 3 hours ago
Xin FuXin Fu
1568
$endgroup$
$begingroup$
Chebyshev might be useful.
$endgroup$
– copper.hat
3 hours ago
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
asked 3 hours ago
Xin FuXin Fu
1568
$endgroup$
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
asked 3 hours ago
Xin FuXin Fu
1568
asked 3 hours ago
Xin FuXin Fu
1568
asked 3 hours ago
Xin FuXin Fu
1568
asked 3 hours ago
Xin FuXin Fu
1568
asked 3 hours ago
Xin FuXin Fu
1568
1568
$begingroup$
Chebyshev might be useful.
$endgroup$
– copper.hat
3 hours ago
add a comment |
$begingroup$
Chebyshev might be useful.
$endgroup$
– copper.hat
3 hours ago
$begingroup$
Chebyshev might be useful.
$endgroup$
– copper.hat
3 hours ago
$begingroup$
Chebyshev might be useful.
$endgroup$
– copper.hat
3 hours ago
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.
answered 3 hours ago
amsmathamsmath
3,364419
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
3 hours ago
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%2fa-problem-in-probability-theory%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.
answered 3 hours ago
amsmathamsmath
3,364419
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
3 hours ago
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.
answered 3 hours ago
amsmathamsmath
3,364419
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
3 hours ago
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.
answered 3 hours ago
amsmathamsmath
3,364419
$endgroup$
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 3 hours ago
amsmathamsmath
3,364419
answered 3 hours ago
amsmathamsmath
3,364419
answered 3 hours ago
amsmathamsmath
3,364419
answered 3 hours ago
amsmathamsmath
3,364419
3,364419
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
3 hours ago
add a comment |
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
3 hours ago
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
3 hours ago
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
3 hours ago
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%2fa-problem-in-probability-theory%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
3 hours ago