A problem in Probability theoryIf $G(x)=P[Xgeq x]$ 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 $mathbbE(X^k+1)$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
How can I get through very long and very dry, but also very useful technical documents when learning a new tool?
Escape a backup date in a file name
Avoiding estate tax by giving multiple gifts
How does Loki do this?
How do I extract a value from a time formatted value in excel?
How to pronounce the slash sign
Unreliable Magic - Is it worth it?
What is paid subscription needed for in Mortal Kombat 11?
How to write papers efficiently when English isn't my first language?
Do the temporary hit points from the Battlerager barbarian's Reckless Abandon stack if I make multiple attacks on my turn?
Was Spock the First Vulcan in Starfleet?
What can we do to stop prior company from asking us questions?
Return the Closest Prime Number
What does 算不上 mean in 算不上太美好的日子?
Lay out the Carpet
What happens if you roll doubles 3 times then land on "Go to jail?"
Increase performance creating Mandelbrot set in python
What is the intuitive meaning of having a linear relationship between the logs of two variables?
Tiptoe or tiphoof? Adjusting words to better fit fantasy races
What is the opposite of 'gravitas'?
A Rare Riley Riddle
Crossing the line between justified force and brutality
Arithmetic mean geometric mean inequality unclear
Is a stroke of luck acceptable after a series of unfavorable events?
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 $mathbbE(X^k+1)$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 $mathbbE(X^2)=1$ and $mathbbE(X)geq a >0$, prove that $$mathbbP(Xgeq lambda a)geq (a-lambda a)^2$$
for $0leqlambda leq 1$.
Let $A=xin Omega:X(x)geq lambda a$, we get
$$int_A (X-lambda a)geq a-int_Alambda a -int_A^cX$$
and $$int_A (X^2-lambda^2 a^2)=1-int_Alambda^2a^2-int_A^cX^2$$
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 $mathbbE(X^2)=1$ and $mathbbE(X)geq a >0$, prove that $$mathbbP(Xgeq lambda a)geq (a-lambda a)^2$$
for $0leqlambda leq 1$.
Let $A=xin Omega:X(x)geq lambda a$, we get
$$int_A (X-lambda a)geq a-int_Alambda a -int_A^cX$$
and $$int_A (X^2-lambda^2 a^2)=1-int_Alambda^2a^2-int_A^cX^2$$
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 $mathbbE(X^2)=1$ and $mathbbE(X)geq a >0$, prove that $$mathbbP(Xgeq lambda a)geq (a-lambda a)^2$$
for $0leqlambda leq 1$.
Let $A=xin Omega:X(x)geq lambda a$, we get
$$int_A (X-lambda a)geq a-int_Alambda a -int_A^cX$$
and $$int_A (X^2-lambda^2 a^2)=1-int_Alambda^2a^2-int_A^cX^2$$
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 $mathbbE(X^2)=1$ and $mathbbE(X)geq a >0$, prove that $$mathbbP(Xgeq lambda a)geq (a-lambda a)^2$$
for $0leqlambda leq 1$.
Let $A=xin Omega:X(x)geq lambda a$, we get
$$int_A (X-lambda a)geq a-int_Alambda a -int_A^cX$$
and $$int_A (X^2-lambda^2 a^2)=1-int_Alambda^2a^2-int_A^cX^2$$
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
$$
alemathbb E(X) = int_Xlelambda aX,dP + int_Xgelambda aX,dP,le,lambda a + int_Xgelambda aX,dP.
$$
Hence,
$$
a(1-lambda),le,int_Xgelambda aX,dP,le,left(int_Xgelambda aX^2,dPright)^1/2cdot P(Xgelambda a)^1/2,le,P(Xgelambda a)^1/2.
$$
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
$$
alemathbb E(X) = int_Xlelambda aX,dP + int_Xgelambda aX,dP,le,lambda a + int_Xgelambda aX,dP.
$$
Hence,
$$
a(1-lambda),le,int_Xgelambda aX,dP,le,left(int_Xgelambda aX^2,dPright)^1/2cdot P(Xgelambda a)^1/2,le,P(Xgelambda a)^1/2.
$$
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
$$
alemathbb E(X) = int_Xlelambda aX,dP + int_Xgelambda aX,dP,le,lambda a + int_Xgelambda aX,dP.
$$
Hence,
$$
a(1-lambda),le,int_Xgelambda aX,dP,le,left(int_Xgelambda aX^2,dPright)^1/2cdot P(Xgelambda a)^1/2,le,P(Xgelambda a)^1/2.
$$
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
$$
alemathbb E(X) = int_Xlelambda aX,dP + int_Xgelambda aX,dP,le,lambda a + int_Xgelambda aX,dP.
$$
Hence,
$$
a(1-lambda),le,int_Xgelambda aX,dP,le,left(int_Xgelambda aX^2,dPright)^1/2cdot P(Xgelambda a)^1/2,le,P(Xgelambda a)^1/2.
$$
Square this and you're done.
answered 3 hours ago
amsmathamsmath
3,364419
$endgroup$
You have
$$
alemathbb E(X) = int_Xlelambda aX,dP + int_Xgelambda aX,dP,le,lambda a + int_Xgelambda aX,dP.
$$
Hence,
$$
a(1-lambda),le,int_Xgelambda aX,dP,le,left(int_Xgelambda aX^2,dPright)^1/2cdot P(Xgelambda a)^1/2,le,P(Xgelambda a)^1/2.
$$
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