Is this a partition of a sample space?Probability on one member of a partition of the sample spacePrecise definition of the support of a random variablePartition probability question - Bayes' Theorem/Law of Total ProbabilityWhat's the sample space for a conditional expectation?Renewal process - sample spaceTotal Probability Theorem / PartitionWhat is the sample space of such random variable?Is it different to sample a random variable $n$ times than it is to sample an $n$-vector once?Is this a Random Variable on the sample space?Measurability of an uncountable union
A sequence that has integer values for prime indexes only:
Is it normal that my co-workers at a fitness company criticize my food choices?
Min function accepting varying number of arguments in C++17
Brexit - No Deal Rejection
How to use deus ex machina safely?
What did Alexander Pope mean by "Expletives their feeble Aid do join"?
What exactly is this small puffer fish doing and how did it manage to accomplish such a feat?
How difficult is it to simply disable/disengage the MCAS on Boeing 737 Max 8 & 9 Aircraft?
Credit cards used everywhere in Singapore or Malaysia?
Why do Australian milk farmers need to protest supermarkets' milk price?
What do Xenomorphs eat in the Alien series?
Have researchers managed to "reverse time"? If so, what does that mean for physics?
Are ETF trackers fundamentally better than individual stocks?
Sailing the cryptic seas
How to write cleanly even if my character uses expletive language?
Recruiter wants very extensive technical details about all of my previous work
How to make healing in an exploration game interesting
Gantt Chart like rectangles with log scale
How could a scammer know the apps on my phone / iTunes account?
How can I track script which gives me "command not found" right after the login?
Do I need to be arrogant to get ahead?
Is it true that good novels will automatically sell themselves on Amazon (and so on) and there is no need for one to waste time promoting?
Why do passenger jet manufacturers design their planes with stall prevention systems?
What is a^b and (a&b)<<1?
Is this a partition of a sample space?
Probability on one member of a partition of the sample spacePrecise definition of the support of a random variablePartition probability question - Bayes' Theorem/Law of Total ProbabilityWhat's the sample space for a conditional expectation?Renewal process - sample spaceTotal Probability Theorem / PartitionWhat is the sample space of such random variable?Is it different to sample a random variable $n$ times than it is to sample an $n$-vector once?Is this a Random Variable on the sample space?Measurability of an uncountable union
$begingroup$
Suppose I have a random variable $ X: Omega longrightarrow R$. Where $Omega$ is the sample space.
Suppose then that I have another random variable $Y$ which is some function of $X$, $Y=f(X)$.
What is the sample space of $Y$? Is it $Omega$ or $R$?
If the sample space is still $Omega$ can I also say that $A_1: X<0 , A_2: X geq 0$ is a partition of $Omega$ ?, or is it only a partition of $R$ since $X$ is real valued?
I am asking this question because I want to apply the law of total expectations on $Y$, i.e. I want to express $E[Y]$ as
$E[Y]=E[Y|X<0]P(X<0) + E[Y|X geq 0]P(X geq 0)$
probability random-variables conditional-expectation expected-value
edited Mar 11 at 23:16
Andrés E. Caicedo
65.7k8160250
asked Mar 11 at 21:33
JesusJesus
11
New contributor
Jesus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Suppose I have a random variable $ X: Omega longrightarrow R$. Where $Omega$ is the sample space.
Suppose then that I have another random variable $Y$ which is some function of $X$, $Y=f(X)$.
What is the sample space of $Y$? Is it $Omega$ or $R$?
If the sample space is still $Omega$ can I also say that $A_1: X<0 , A_2: X geq 0$ is a partition of $Omega$ ?, or is it only a partition of $R$ since $X$ is real valued?
I am asking this question because I want to apply the law of total expectations on $Y$, i.e. I want to express $E[Y]$ as
$E[Y]=E[Y|X<0]P(X<0) + E[Y|X geq 0]P(X geq 0)$
probability random-variables conditional-expectation expected-value
edited Mar 11 at 23:16
Andrés E. Caicedo
65.7k8160250
asked Mar 11 at 21:33
JesusJesus
11
New contributor
Jesus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Suppose I have a random variable $ X: Omega longrightarrow R$. Where $Omega$ is the sample space.
Suppose then that I have another random variable $Y$ which is some function of $X$, $Y=f(X)$.
What is the sample space of $Y$? Is it $Omega$ or $R$?
If the sample space is still $Omega$ can I also say that $A_1: X<0 , A_2: X geq 0$ is a partition of $Omega$ ?, or is it only a partition of $R$ since $X$ is real valued?
I am asking this question because I want to apply the law of total expectations on $Y$, i.e. I want to express $E[Y]$ as
$E[Y]=E[Y|X<0]P(X<0) + E[Y|X geq 0]P(X geq 0)$
probability random-variables conditional-expectation expected-value
edited Mar 11 at 23:16
Andrés E. Caicedo
65.7k8160250
asked Mar 11 at 21:33
JesusJesus
11
New contributor
Jesus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Suppose I have a random variable $ X: Omega longrightarrow R$. Where $Omega$ is the sample space.
Suppose then that I have another random variable $Y$ which is some function of $X$, $Y=f(X)$.
What is the sample space of $Y$? Is it $Omega$ or $R$?
If the sample space is still $Omega$ can I also say that $A_1: X<0 , A_2: X geq 0$ is a partition of $Omega$ ?, or is it only a partition of $R$ since $X$ is real valued?
I am asking this question because I want to apply the law of total expectations on $Y$, i.e. I want to express $E[Y]$ as
$E[Y]=E[Y|X<0]P(X<0) + E[Y|X geq 0]P(X geq 0)$
probability random-variables conditional-expectation expected-value
probability random-variables conditional-expectation expected-value
edited Mar 11 at 23:16
Andrés E. Caicedo
65.7k8160250
asked Mar 11 at 21:33
JesusJesus
11
New contributor
Jesus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Mar 11 at 23:16
Andrés E. Caicedo
65.7k8160250
asked Mar 11 at 21:33
JesusJesus
11
New contributor
Jesus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Mar 11 at 23:16
Andrés E. Caicedo
65.7k8160250
edited Mar 11 at 23:16
Andrés E. Caicedo
65.7k8160250
edited Mar 11 at 23:16
Andrés E. Caicedo
65.7k8160250
65.7k8160250
asked Mar 11 at 21:33
JesusJesus
11
New contributor
Jesus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Mar 11 at 21:33
JesusJesus
11
asked Mar 11 at 21:33
JesusJesus
11
11
New contributor
Jesus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Jesus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Jesus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The sample space is normally still $Omega$ in that $Y:=fcirc X :Omega to mathbbR$. Furthermore $A_1$ and $A_2$ are disjoint subsets of $Omega$ in that
$$
A_1 = (X<0) = X^-1((-infty,0)) = omegainOmega: X(omega)<0 subseteq Omega,
$$
and
$$
A_1 = (Xgeq 0) =X^-1([0,infty)) = omegainOmega: X(omega)geq 0 subseteq Omega,
$$
for which it holds that
$$
A_1cap A_2 = omegainOmega:X(omega)<0, X(omega)geq 0 = emptyset.
$$
They also cover the entire of our sample space, that is
$$
A_1 cup A_2 = omegainOmega: X(omega)in mathbbR = Omega.
$$
Thus this is a partition of $Omega$ if we don't require a partition not to include the empty set. Otherwise you have to include more information about $X$ not being strictly negative or non-negative.
To use the formula you use you need $P(A_1),P(A_2)>0$, implying that they indeed are non-empty. However given they have positive probability you can indeed use the law of total expectation.
answered Mar 11 at 22:22
MartinMartin
971717
$endgroup$
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
);
);
Jesus is a new contributor. Be nice, and check out our Code of Conduct.
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%2f3144284%2fis-this-a-partition-of-a-sample-space%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
$begingroup$
The sample space is normally still $Omega$ in that $Y:=fcirc X :Omega to mathbbR$. Furthermore $A_1$ and $A_2$ are disjoint subsets of $Omega$ in that
$$
A_1 = (X<0) = X^-1((-infty,0)) = omegainOmega: X(omega)<0 subseteq Omega,
$$
and
$$
A_1 = (Xgeq 0) =X^-1([0,infty)) = omegainOmega: X(omega)geq 0 subseteq Omega,
$$
for which it holds that
$$
A_1cap A_2 = omegainOmega:X(omega)<0, X(omega)geq 0 = emptyset.
$$
They also cover the entire of our sample space, that is
$$
A_1 cup A_2 = omegainOmega: X(omega)in mathbbR = Omega.
$$
Thus this is a partition of $Omega$ if we don't require a partition not to include the empty set. Otherwise you have to include more information about $X$ not being strictly negative or non-negative.
To use the formula you use you need $P(A_1),P(A_2)>0$, implying that they indeed are non-empty. However given they have positive probability you can indeed use the law of total expectation.
answered Mar 11 at 22:22
MartinMartin
971717
$endgroup$
add a comment |
$begingroup$
The sample space is normally still $Omega$ in that $Y:=fcirc X :Omega to mathbbR$. Furthermore $A_1$ and $A_2$ are disjoint subsets of $Omega$ in that
$$
A_1 = (X<0) = X^-1((-infty,0)) = omegainOmega: X(omega)<0 subseteq Omega,
$$
and
$$
A_1 = (Xgeq 0) =X^-1([0,infty)) = omegainOmega: X(omega)geq 0 subseteq Omega,
$$
for which it holds that
$$
A_1cap A_2 = omegainOmega:X(omega)<0, X(omega)geq 0 = emptyset.
$$
They also cover the entire of our sample space, that is
$$
A_1 cup A_2 = omegainOmega: X(omega)in mathbbR = Omega.
$$
Thus this is a partition of $Omega$ if we don't require a partition not to include the empty set. Otherwise you have to include more information about $X$ not being strictly negative or non-negative.
To use the formula you use you need $P(A_1),P(A_2)>0$, implying that they indeed are non-empty. However given they have positive probability you can indeed use the law of total expectation.
answered Mar 11 at 22:22
MartinMartin
971717
$endgroup$
add a comment |
$begingroup$
The sample space is normally still $Omega$ in that $Y:=fcirc X :Omega to mathbbR$. Furthermore $A_1$ and $A_2$ are disjoint subsets of $Omega$ in that
$$
A_1 = (X<0) = X^-1((-infty,0)) = omegainOmega: X(omega)<0 subseteq Omega,
$$
and
$$
A_1 = (Xgeq 0) =X^-1([0,infty)) = omegainOmega: X(omega)geq 0 subseteq Omega,
$$
for which it holds that
$$
A_1cap A_2 = omegainOmega:X(omega)<0, X(omega)geq 0 = emptyset.
$$
They also cover the entire of our sample space, that is
$$
A_1 cup A_2 = omegainOmega: X(omega)in mathbbR = Omega.
$$
Thus this is a partition of $Omega$ if we don't require a partition not to include the empty set. Otherwise you have to include more information about $X$ not being strictly negative or non-negative.
To use the formula you use you need $P(A_1),P(A_2)>0$, implying that they indeed are non-empty. However given they have positive probability you can indeed use the law of total expectation.
answered Mar 11 at 22:22
MartinMartin
971717
$endgroup$
The sample space is normally still $Omega$ in that $Y:=fcirc X :Omega to mathbbR$. Furthermore $A_1$ and $A_2$ are disjoint subsets of $Omega$ in that
$$
A_1 = (X<0) = X^-1((-infty,0)) = omegainOmega: X(omega)<0 subseteq Omega,
$$
and
$$
A_1 = (Xgeq 0) =X^-1([0,infty)) = omegainOmega: X(omega)geq 0 subseteq Omega,
$$
for which it holds that
$$
A_1cap A_2 = omegainOmega:X(omega)<0, X(omega)geq 0 = emptyset.
$$
They also cover the entire of our sample space, that is
$$
A_1 cup A_2 = omegainOmega: X(omega)in mathbbR = Omega.
$$
Thus this is a partition of $Omega$ if we don't require a partition not to include the empty set. Otherwise you have to include more information about $X$ not being strictly negative or non-negative.
To use the formula you use you need $P(A_1),P(A_2)>0$, implying that they indeed are non-empty. However given they have positive probability you can indeed use the law of total expectation.
answered Mar 11 at 22:22
MartinMartin
971717
answered Mar 11 at 22:22
MartinMartin
971717
answered Mar 11 at 22:22
MartinMartin
971717
answered Mar 11 at 22:22
MartinMartin
971717
971717
add a comment |
add a comment |
Jesus is a new contributor. Be nice, and check out our Code of Conduct.
Jesus is a new contributor. Be nice, and check out our Code of Conduct.
Jesus is a new contributor. Be nice, and check out our Code of Conduct.
Jesus is a new contributor. Be nice, and check out our Code of Conduct.
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%2f3144284%2fis-this-a-partition-of-a-sample-space%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
