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

Multi tool use
Multi tool use

Would this custom Sorcerer variant that can only learn any verbal-component-only spell be unbalanced?

What is the intuitive meaning of having a linear relationship between the logs of two variables?

India just shot down a satellite from the ground. At what altitude range is the resulting debris field?

Is a stroke of luck acceptable after a series of unfavorable events?

How easy is it to start Magic from scratch?

Is oxalic acid dihydrate considered a primary acid standard in analytical chemistry?

How did Doctor Strange see the winning outcome in Avengers: Infinity War?

Is there a good way to store credentials outside of a password manager?

What does "I’d sit this one out, Cap," imply or mean in the context?

Replace character with another only if repeated and not part of a word

What happens if you roll doubles 3 times then land on "Go to jail?"

Hostile work environment after whistle-blowing on coworker and our boss. What do I do?

Why Were Madagascar and New Zealand Discovered So Late?

Is the destination of a commercial flight important for the pilot?

How does buying out courses with grant money work?

How can a function with a hole (removable discontinuity) equal a function with no hole?

Applicability of Single Responsibility Principle

What is the best translation for "slot" in the context of multiplayer video games?

What is the opposite of 'gravitas'?

Tiptoe or tiphoof? Adjusting words to better fit fantasy races

Was a professor correct to chastise me for writing "Dear Prof. X" rather than "Dear Professor X"?

How do scammers retract money, while you can’t?

Did Dumbledore lie to Harry about how long he had James Potter's invisibility cloak when he was examining it? If so, why?

Would a high gravity rocky planet be guaranteed to have an atmosphere?



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













4












$begingroup$


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







share|cite|improve this question









$endgroup$











  • $begingroup$
    Chebyshev might be useful.
    $endgroup$
    – copper.hat
    2 hours ago















4












$begingroup$


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







share|cite|improve this question









$endgroup$











  • $begingroup$
    Chebyshev might be useful.
    $endgroup$
    – copper.hat
    2 hours ago













4












4








4


1



$begingroup$


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







share|cite|improve this question









$endgroup$




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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 3 hours ago









Xin FuXin Fu

1568




1568











  • $begingroup$
    Chebyshev might be useful.
    $endgroup$
    – copper.hat
    2 hours ago
















  • $begingroup$
    Chebyshev might be useful.
    $endgroup$
    – copper.hat
    2 hours ago















$begingroup$
Chebyshev might be useful.
$endgroup$
– copper.hat
2 hours ago




$begingroup$
Chebyshev might be useful.
$endgroup$
– copper.hat
2 hours ago










1 Answer
1






active

oldest

votes


















5












$begingroup$

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.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thank you very much!
    $endgroup$
    – Xin Fu
    2 hours ago










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
);



);













draft saved

draft discarded


















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









5












$begingroup$

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.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thank you very much!
    $endgroup$
    – Xin Fu
    2 hours ago















5












$begingroup$

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.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thank you very much!
    $endgroup$
    – Xin Fu
    2 hours ago













5












5








5





$begingroup$

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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 2 hours ago









amsmathamsmath

3,364419




3,364419











  • $begingroup$
    Thank you very much!
    $endgroup$
    – Xin Fu
    2 hours ago
















  • $begingroup$
    Thank you very much!
    $endgroup$
    – Xin Fu
    2 hours ago















$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
2 hours ago




$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
2 hours ago

















draft saved

draft discarded
















































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.




draft saved


draft discarded














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





















































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







wFPg,DPjNqVKdeDMuZcmnGnY3C,4ZYde XfkdyGz,0zfh 7
-
KvZgf,RmHGFz5i aYpGJ,lekkrdQf,JpIx6Vmiqqfoe04IUE3jEFLY,wGhhr,ixTmRmo nK5FXId,lS1Jqgj,uVC43VSbM

Popular posts from this blog

Creating centerline of river in QGIS? The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Finding centrelines from polygons in QGIS?Splitting line into two lines with GRASS GIS?Centroid of the equator and a pointpostgis: problems creating flow direction polyline; not all needed connections are drawnhow to make decent sense from scattered river depth measurementsQGIS Interpolation on Curved Grid (River DEMs)How to create automatic parking baysShortest path creation between two linesclipping layer using query builder in QGISFinding which side of closest polyline point lies on in QGIS?Create centerline from multi-digitized roadway lines Qgis 2.18Getting bathymetric contours confined only within river banks using QGIS?

Image
Calculating the number of coins in money change How could chocolate be as addictive as heroin? What do you call a plan that's an alternative plan in case your initial plan fails? Python - Fishing Simulator Simulating Exploding Dice Active filter with series inductor and resistor - do these exist? How can I protect witches in combat who wear limited clothing? Am I ethically obligated to go into work on an off day if the reason is sudden? Format single node in tikzcd Didn't get enough time to take a Coding Test - what to do now? Why can't wing-mounted spoilers be used to steepen approaches? How is simplicity better than precision and clarity in prose? Is there a trick to getting spices to fix to nuts? What can I do if neighbor is blocking my solar panels intentionally? Hiding Certain Lines on Table Wolves and sheep Make it rain characters Can the prologue be the backstory of your main character? Difference between はじめる and ...
Read more

What is the result of assigning to std::vector::begin()? The Next CEO of Stack OverflowWhat are the differences between a pointer variable and a reference variable in C++?What does the explicit keyword mean?Concatenating two std::vectorsHow to find out if an item is present in a std::vector?Why is “using namespace std” considered bad practice?What is the “-->” operator in C++?What is the easiest way to initialize a std::vector with hardcoded elements?What is The Rule of Three?What are the basic rules and idioms for operator overloading?Why are std::begin and std::end “not memory safe”?

Image
Is there a way to save my career from absolute disaster? Why didn't Khan get resurrected in the Genesis Explosion? How to make a variable always equal to the result of some calculations? What can we do to stop prior company from asking us questions? How do we know the LHC results are robust? How does the mv command work with external drives? What is ( CFMCC ) on ILS approach chart? Between two walls Should I tutor a student who I know has cheated on their homework? How do scammers retract money, while you can’t? What happens if you roll doubles 3 times then land on "Go to jail?" What is the result of assigning to std::vector<T>::begin()? How to count occurrences of text in a file? What was the first Unix version to run on a microcomputer? Return the Closest Prime Number How did people program for Consoles with multiple CPUs? What does convergence in distribution "in the Gromov–Hausdorff" sense mean? How should I support th...
Read more