$X$ 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
Which one is the true statement?
Audio Conversion With ADS1243
Is there such a thing as a proper verb, like a proper noun?
What steps are necessary to read a Modern SSD in Medieval Europe?
Where do students learn to solve polynomial equations these days?
Does higher Oxidation/ reduction potential translate to higher energy storage in battery?
What happened in Rome, when the western empire "fell"?
free fall ellipse or parabola?
Help understanding this unsettling image of Titan, Epimetheus, and Saturn's rings?
Is it ok to trim down a tube patch?
Is there an equivalent of cd - for cp or mv
Which Pokemon have a special animation when running with them out of their pokeball?
Do scriptures give a method to recognize a truly self-realized person/jivanmukta?
Players Circumventing the limitations of Wish
Point distance program written without a framework
(How) Could a medieval fantasy world survive a magic-induced "nuclear winter"?
How to use ReplaceAll on an expression that contains a rule
Is a distribution that is normal, but highly skewed, considered Gaussian?
What does "shotgun unity" refer to here in this sentence?
Is it ever safe to open a suspicious HTML file (e.g. email attachment)?
"Eavesdropping" vs "Listen in on"
Do I need to write [sic] when including a quotation with a number less than 10 that isn't written out?
Getting Stale Gas Out of a Gas Tank w/out Dropping the Tank
Why don't programming languages automatically manage the synchronous/asynchronous problem?
$X$ 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
$begingroup$
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
edited Mar 28 at 9:19
Asaf Karagila♦
307k33440773
asked Mar 28 at 3:11
Xin FuXin Fu
33319
$endgroup$
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
edited Mar 28 at 9:19
Asaf Karagila♦
307k33440773
asked Mar 28 at 3:11
Xin FuXin Fu
33319
$endgroup$
$begingroup$
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 $mathbbE(X^2)=1$ 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
edited Mar 28 at 9:19
Asaf Karagila♦
307k33440773
asked Mar 28 at 3:11
Xin FuXin Fu
33319
$endgroup$
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
edited Mar 28 at 9:19
Asaf Karagila♦
307k33440773
asked Mar 28 at 3:11
Xin FuXin Fu
33319
edited Mar 28 at 9:19
Asaf Karagila♦
307k33440773
asked Mar 28 at 3:11
Xin FuXin Fu
33319
edited Mar 28 at 9:19
Asaf Karagila♦
307k33440773
edited Mar 28 at 9:19
Asaf Karagila♦
307k33440773
edited Mar 28 at 9:19
Asaf Karagila♦
307k33440773
307k33440773
asked Mar 28 at 3:11
Xin FuXin Fu
33319
asked Mar 28 at 3:11
Xin FuXin Fu
33319
asked Mar 28 at 3:11
Xin FuXin Fu
33319
33319
$begingroup$
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
$$
alemathbb E(X) = int_Xlelambda aX,dP + int_Xgelambda aX,dP,le,lambda a + int_Xgelambda aX,dP.
$$
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 Mar 28 at 3:37
amsmathamsmath
3,288420
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
Mar 28 at 3:44
add a comment |
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
$$
alemathbb E(X) = int_Xlelambda aX,dP + int_Xgelambda aX,dP,le,lambda a + int_Xgelambda aX,dP.
$$
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 Mar 28 at 3:37
amsmathamsmath
3,288420
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
Mar 28 at 3:44
add a comment |
$begingroup$
You have
$$
alemathbb E(X) = int_Xlelambda aX,dP + int_Xgelambda aX,dP,le,lambda a + int_Xgelambda aX,dP.
$$
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 Mar 28 at 3:37
amsmathamsmath
3,288420
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– Xin Fu
Mar 28 at 3:44
add a comment |
$begingroup$
You have
$$
alemathbb E(X) = int_Xlelambda aX,dP + int_Xgelambda aX,dP,le,lambda a + int_Xgelambda aX,dP.
$$
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 Mar 28 at 3:37
amsmathamsmath
3,288420
$endgroup$
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 Mar 28 at 3:37
amsmathamsmath
3,288420
answered Mar 28 at 3:37
amsmathamsmath
3,288420
answered Mar 28 at 3:37
amsmathamsmath
3,288420
answered Mar 28 at 3:37
amsmathamsmath
3,288420
3,288420
$begingroup$
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