$X$ is a random variable, if $Bbb E(X^2)=1$ and $Bbb E(X)geq a>0$, prove that $Bbb P(Xgeqlambda a)geq(a-lambda a)^2$ for $0leqlambdaleq 1$. The Next CEO of Stack OverflowHow to show $P[Xgeq k] leq (fraclambda ek)^k e^-lambda$ (X is Poisson random variable)Show that $P(X > lambda) geq frac(EX - lambda)^2EX^2$An equivalent condition for a random variable to be integrableProof that absolute value of a random variable is a random variableIs PDF unique for a random variable $X$ in given probability space?Any technique to show that $PleftXgeq 2lambdarightleq (e/4)^lambda$Is this a valid identity for the expected value of a nonnegative random variable?Convergence of $y^q P(X>y)$ for $y to infty$ , $X geq 0$ random variable, $q>0$, $mathbbE[X^q]<1$Explicitly representing a random variable such as $ X(omega):=frac1lambda ln frac11-omega$, which is exponentialNotation for expectation of a random variable/general Lebesgue integral

How to safely derail a train during transit?

Why is there a PLL in CPU?

How to write papers efficiently when English isn't my first language?

Increase performance creating Mandelbrot set in python

How to write the block matrix in LaTex?

How do I construct this japanese bowl?

What does "Its cash flow is deeply negative" mean?

How do I go from 300 unfinished/half written blog posts, to published posts?

Describing a person. What needs to be mentioned?

MAZDA 3 2006 (UK) - poor acceleration then takes off at 3250 revs

How to make a variable always equal to the result of some calculations?

Can a single photon have an energy density?

Is it a good idea to use COLUMN AS (left([Another_Column],(4)) instead of LEFT in the select?

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

How can I quit an app using Terminal?

Why do professional authors make "consistency" mistakes? And how to avoid them?

Why didn't Khan get resurrected in the Genesis Explosion?

Anatomically Correct Mesopelagic Aves

If I blow insulation everywhere in my attic except the door trap, will heat escape through it?

How do spells that require an ability check vs. the caster's spell save DC work?

What makes a siege story/plot interesting?

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

Why doesn't a table tennis ball float on the surface? How do we calculate buoyancy here?

Putting a 2D region plot under a 3D plot



$X$ is a random variable, if $Bbb E(X^2)=1$ and $Bbb E(X)geq a>0$, prove that $Bbb P(Xgeqlambda a)geq(a-lambda a)^2$ for $0leqlambdaleq 1$.



The Next CEO of Stack OverflowHow to show $P[Xgeq k] leq (fraclambda ek)^k e^-lambda$ (X is Poisson random variable)Show that $P(X > lambda) geq frac(EX - lambda)^2EX^2$An equivalent condition for a random variable to be integrableProof that absolute value of a random variable is a random variableIs PDF unique for a random variable $X$ in given probability space?Any technique to show that $PleftXgeq 2lambdarightleq (e/4)^lambda$Is this a valid identity for the expected value of a nonnegative random variable?Convergence of $y^q P(X>y)$ for $y to infty$ , $X geq 0$ random variable, $q>0$, $mathbbE[X^q]<1$Explicitly representing a random variable such as $ X(omega):=frac1lambda ln frac11-omega$, which is exponentialNotation for expectation of a random variable/general Lebesgue integral










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
    22 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
    22 hours ago













4












4








4


2



$begingroup$


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








edited 16 hours ago









Asaf Karagila♦

307k33439771




307k33439771










asked 22 hours ago









Xin FuXin Fu

1618




1618











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
















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















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




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










1 Answer
1






active

oldest

votes


















6












$begingroup$

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
    21 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%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









6












$begingroup$

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
    21 hours ago















6












$begingroup$

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
    21 hours ago













6












6








6





$begingroup$

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 22 hours ago









amsmathamsmath

3,263419




3,263419











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
















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















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




$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
21 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%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





















































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







-

Popular posts from this blog

RemoteApp sporadic failureWindows 2008 RemoteAPP client disconnects within a matter of minutesWhat is the minimum version of RDP supported by Server 2012 RDS?How to configure a Remoteapp server to increase stabilityMicrosoft RemoteApp Active SessionRDWeb TS connection broken for some users post RemoteApp certificate changeRemote Desktop Licensing, RemoteAPPRDS 2012 R2 some users are not able to logon after changed date and time on Connection BrokersWhat happens during Remote Desktop logon, and is there any logging?After installing RDS on WinServer 2016 I still can only connect with two users?RD Connection via RDGW to Session host is not connecting

Image
Can I use my laptop, which says 100-240V, in the USA? Definition of Newton's first law Why did the ICC decide not to probe alleged US atrocities in Afghanistan? Is there ever any indication in the MCU as to how Spider-Man got his powers? Should pressure only be applied at the top of a pastry bag? How did Thanos not realise this had happened at the end of Endgame? On studying Computer Science vs. Software Engineering to become a proficient coder Why does the Earth follow an elliptical trajectory rather than a parabolic one? What's the word for the soldier salute? Why not just directly invest in the holdings of an ETF? Can a tourist shoot a gun in the USA? Why in a Ethernet LAN, a packet sniffer can obtain all packets sent over the LAN? What is Plautus’s pun about frustum and frustrum? Extrude the faces of a cube symmetrically along XYZ Why does my circuit work on a breadboard, but not on a perfboard? I am new to soldering How do I tell my supervisor...
Read more

How to write a 12-bar blues melodyI-IV-V blues progressionHow to play the bridges in a standard blues progressionHow does Gdim7 fit in C# minor?question on a certain chord progressionMusicology of Melody12 bar blues, spread rhythm: alternative to 6th chord to avoid finger stretchChord progressions/ Root key/ MelodiesHow to put chords (POP-EDM) under a given lead vocal melody (starting from a good knowledge in music theory)Are there “rules” for improvising with the minor pentatonic scale over 12-bar shuffle?Confusion about blues scale and chords

Image
Why is a set not a partition of itself? Would an 8% reduction in drag outweigh the weight addition from this custom CFD-tested winglet? Labeling matrices/rectangles and drawing Sigma inside rectangle Unexpected Netflix account registered to my Gmail address - any way it could be a hack attempt? 51% attack - apparently very easy? refering to CZ's "rollback btc chain" - How to make sure such corruptible scenario can never happen so easily? Missouri raptors have wild hairdos Can someone explain homicide-related death rates? Why is tomato paste so cheap? What are the implications of the new alleged key recovery attack preprint on SIMON? Effects of ~10atm pressure on engine design Is taking modulus on both sides of an equation valid? Jumping frame contents with beamer and pgfplots Are there any established rules for splitting books into parts, chapters, sections etc? Anatomically Correct Carnivorous Tree Why do I get two different answers when solvi...
Read more

Esgonzo ibérico Índice Descrición Distribución Hábitat Ameazas Notas Véxase tamén "Acerca dos nomes dos anfibios e réptiles galegos""Chalcides bedriagai"Chalcides bedriagai en Carrascal, L. M. Salvador, A. (Eds). Enciclopedia virtual de los vertebrados españoles. Museo Nacional de Ciencias Naturales, Madrid. España.Fotos

Image
escíncidospenínsula Ibéricaesgonzo comúnJacques von Bedriagaesgonzo comúndimorfismo sexualovovivíparaendémicaPenínsula IbéricaCordilleira CantábricaAsturiasPaís VascoGaliciaRías BaixasMurosCarnotaillas CíesOnsAtlánticoIllas CíesIlla de Onsilla do PesegueiroIlla de Sancti PetriMediterráneoMar MenorIlla de TabarcaGaliciaCantabriaBurgossolosdestrución do hábitatIUCN Abrir o menú principal Procurar Esgonzo ibérico Ler noutra lingua Vixiar esta páxina Editar function mfTempOpenSection(id)var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block"; Esgonzo ibérico Estado de conservación Case ameazada Clasificaci...
Read more