Dtjytyk

Can I say "fingers" when referring to toes?

How would you translate "more" for use as an interface button?

What does "Scientists rise up against statistical significance" mean? (Comment in Nature)

Change the color of a single dot in `ddot` symbol

Why does Carol not get rid of the Kree symbol on her suit when she changes its colours?

Did the UK lift the requirement for registering SIM cards?

Delete multiple columns using awk or sed

Does the reader need to like the PoV character?

How to get directions in deep space?

What features enable the Su-25 Frogfoot to operate with such a wide variety of fuels?

Does "he squandered his car on drink" sound natural?

What is the highest possible scrabble score for placing a single tile

Is there a RAID 0 Equivalent for RAM?

How does electrical safety system work on ISS?

Which Article Helped Get Rid of Technobabble in RPGs?

How to make money from a browser who sees 5 seconds into the future of any web page?

Has any country ever had 2 former presidents in jail simultaneously?

How to preserve electronics (computers, iPads and phones) for hundreds of years

Strong empirical falsification of quantum mechanics based on vacuum energy density?

Make a Bowl of Alphabet Soup

Taxes on Dividends in a Roth IRA

Doesn't the system of the Supreme Court oppose justice?

Non-trope happy ending?

Is this toilet slogan correct usage of the English language?

Counter-example to “Every RV has a CDF.”

Form of $sigma(X_n)$ and $sigma(X_n)$-measurabilityIs a predictable process adapted?Convergence in probability to a non-measurable limitUnderstanding this Dynkin system $pi-lambda$ proof that a CDF completely determines a distribution?Measurability of randomly chosen coordinateHelp validating counter-example (basic probability - generated $sigma$-algebra)Making sense of measure-theoretic definition of random variableWhy is the canonical product $sigma$-algebra the right $sigma$-algebra on the product space?IID discrete but not a.s. equalHitting time in general not measurable

2

$begingroup$

I’ve seen the statement that all RVs have CDFs and vice versa in many textbooks.
The way they show is by simply constructing $F_X(x) = P(X in (infty, x))$.

But that assumes $forall xinmathbb R: (infty, x) in mathcal F$, which isn’t necessarily true for all $sigma$-algebras on $mathbb R$. Thus, haven’t we found a counter-example?

To be explicit, the definition of RVs I’m using is: measurable functions from the sample space of a probability space $(Omega, mathcal F, P)$ to any measurable space $(A, mathcal A)$.

probability-theory

share|cite|improve this question

edited Mar 13 at 8:17

Yatharth Agarwal

asked Mar 13 at 8:00

Yatharth AgarwalYatharth Agarwal

542418

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  There are many similarly-titled questions, but imo this is not a duplicate. I’m asking about a specific counter-example. I could try to edit the title to make it sound more distinct if needed.
  $endgroup$
  – Yatharth Agarwal
  Mar 13 at 8:01
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The term 'random variable' has a fixed universally accepted meaning and what you are trying to do is to change its meaning.
  $endgroup$
  – Kavi Rama Murthy
  Mar 13 at 8:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @KaviRamaMurthy I think you’re definitely right in that we’re working with different definitions of RVs. I edited the question to state mine. Could you state yours? I know you mentioned it’s a universally accepted meaning, but somehow I seem to have missed it in the references I consulted.
  $endgroup$
  – Yatharth Agarwal
  Mar 13 at 8:10
  

  
  
  
  

add a comment | 

2

$begingroup$

I’ve seen the statement that all RVs have CDFs and vice versa in many textbooks.
The way they show is by simply constructing $F_X(x) = P(X in (infty, x))$.

But that assumes $forall xinmathbb R: (infty, x) in mathcal F$, which isn’t necessarily true for all $sigma$-algebras on $mathbb R$. Thus, haven’t we found a counter-example?

To be explicit, the definition of RVs I’m using is: measurable functions from the sample space of a probability space $(Omega, mathcal F, P)$ to any measurable space $(A, mathcal A)$.

probability-theory

share|cite|improve this question

edited Mar 13 at 8:17

Yatharth Agarwal

asked Mar 13 at 8:00

Yatharth AgarwalYatharth Agarwal

542418

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  There are many similarly-titled questions, but imo this is not a duplicate. I’m asking about a specific counter-example. I could try to edit the title to make it sound more distinct if needed.
  $endgroup$
  – Yatharth Agarwal
  Mar 13 at 8:01
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The term 'random variable' has a fixed universally accepted meaning and what you are trying to do is to change its meaning.
  $endgroup$
  – Kavi Rama Murthy
  Mar 13 at 8:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @KaviRamaMurthy I think you’re definitely right in that we’re working with different definitions of RVs. I edited the question to state mine. Could you state yours? I know you mentioned it’s a universally accepted meaning, but somehow I seem to have missed it in the references I consulted.
  $endgroup$
  – Yatharth Agarwal
  Mar 13 at 8:10
  

  
  
  
  

add a comment | 

$begingroup$

I’ve seen the statement that all RVs have CDFs and vice versa in many textbooks.
The way they show is by simply constructing $F_X(x) = P(X in (infty, x))$.

But that assumes $forall xinmathbb R: (infty, x) in mathcal F$, which isn’t necessarily true for all $sigma$-algebras on $mathbb R$. Thus, haven’t we found a counter-example?

To be explicit, the definition of RVs I’m using is: measurable functions from the sample space of a probability space $(Omega, mathcal F, P)$ to any measurable space $(A, mathcal A)$.

probability-theory

share|cite|improve this question

edited Mar 13 at 8:17

Yatharth Agarwal

asked Mar 13 at 8:00

Yatharth AgarwalYatharth Agarwal

542418

$endgroup$

share|cite|improve this question

edited Mar 13 at 8:17

Yatharth Agarwal

asked Mar 13 at 8:00

Yatharth AgarwalYatharth Agarwal

542418

share|cite|improve this question

edited Mar 13 at 8:17

Yatharth Agarwal

asked Mar 13 at 8:00

Yatharth AgarwalYatharth Agarwal

542418

asked Mar 13 at 8:00

Yatharth AgarwalYatharth Agarwal

542418

asked Mar 13 at 8:00

Yatharth AgarwalYatharth Agarwal

542418

1 Answer
1

active

oldest

votes

1

$begingroup$

What you have defined is often called a random element of $A$. In contrast, random variables are usually defined as measurable functions from a probability space to $(mathbb R, mathcal B(mathbb R))$. With this restriction, the definition $F_X(x)=P(Xleq x)$ is always valid.

If a random variable is defined as measurable map from a measure space into a measurable space the we can associate with it what is called the induced measure: $nu (A)=P(X^{-1}(A))$ for $A in mathcal A$. Sometimes this is also called the distribution of $X$. It so happens that in the case $A=mathbb R$ with Borel sigma algebra there is a one-to-one correspondence between induced measures and the functions $F_X$.

share|cite|improve this answer

edited Mar 13 at 8:22

Yatharth Agarwal

542418

answered Mar 13 at 8:04

Kavi Rama MurthyKavi Rama Murthy

68.9k53169

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hmm… the definitions I’ve seen only required $X$ map to a measurable space $(A, mathcal A)$, not necessarily $A = mathbb R$ and not necessarily $mathcal A = mathcal B (mathbb R)$.
  $endgroup$
  – Yatharth Agarwal
  Mar 13 at 8:07
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you provide a reference? I’d love to dig deeper.
  $endgroup$
  – Yatharth Agarwal
  Mar 13 at 8:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  In that case you are asking about existence of what? Can you give a precise definition of the term 'random variable' and the term 'distribution function'?
  $endgroup$
  – Kavi Rama Murthy
  Mar 13 at 8:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @YatharthAgarwal What you have defined is often called a random element of $A$. In any case please see if my revised answer is clear.
  $endgroup$
  – Kavi Rama Murthy
  Mar 13 at 8:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ahhh random elements vs random variables. The text I was using didn’t distinguish. This makes sense—thank you!
  $endgroup$
  – Yatharth Agarwal
  Mar 13 at 8:19
  

  
  
  
  

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
});


}
});

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3146249%2fcounter-example-to-every-rv-has-a-cdf%23new-answer', 'question_page');
}
);

Post as a guest

Name

Email

Required, but never shown

1

$begingroup$

What you have defined is often called a random element of $A$. In contrast, random variables are usually defined as measurable functions from a probability space to $(mathbb R, mathcal B(mathbb R))$. With this restriction, the definition $F_X(x)=P(Xleq x)$ is always valid.

If a random variable is defined as measurable map from a measure space into a measurable space the we can associate with it what is called the induced measure: $nu (A)=P(X^{-1}(A))$ for $A in mathcal A$. Sometimes this is also called the distribution of $X$. It so happens that in the case $A=mathbb R$ with Borel sigma algebra there is a one-to-one correspondence between induced measures and the functions $F_X$.

share|cite|improve this answer

edited Mar 13 at 8:22

Yatharth Agarwal

542418

answered Mar 13 at 8:04

Kavi Rama MurthyKavi Rama Murthy

68.9k53169

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hmm… the definitions I’ve seen only required $X$ map to a measurable space $(A, mathcal A)$, not necessarily $A = mathbb R$ and not necessarily $mathcal A = mathcal B (mathbb R)$.
  $endgroup$
  – Yatharth Agarwal
  Mar 13 at 8:07
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you provide a reference? I’d love to dig deeper.
  $endgroup$
  – Yatharth Agarwal
  Mar 13 at 8:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  In that case you are asking about existence of what? Can you give a precise definition of the term 'random variable' and the term 'distribution function'?
  $endgroup$
  – Kavi Rama Murthy
  Mar 13 at 8:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @YatharthAgarwal What you have defined is often called a random element of $A$. In any case please see if my revised answer is clear.
  $endgroup$
  – Kavi Rama Murthy
  Mar 13 at 8:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ahhh random elements vs random variables. The text I was using didn’t distinguish. This makes sense—thank you!
  $endgroup$
  – Yatharth Agarwal
  Mar 13 at 8:19
  

  
  
  
  

add a comment | 

1

$begingroup$

What you have defined is often called a random element of $A$. In contrast, random variables are usually defined as measurable functions from a probability space to $(mathbb R, mathcal B(mathbb R))$. With this restriction, the definition $F_X(x)=P(Xleq x)$ is always valid.

If a random variable is defined as measurable map from a measure space into a measurable space the we can associate with it what is called the induced measure: $nu (A)=P(X^{-1}(A))$ for $A in mathcal A$. Sometimes this is also called the distribution of $X$. It so happens that in the case $A=mathbb R$ with Borel sigma algebra there is a one-to-one correspondence between induced measures and the functions $F_X$.

share|cite|improve this answer

edited Mar 13 at 8:22

Yatharth Agarwal

542418

answered Mar 13 at 8:04

Kavi Rama MurthyKavi Rama Murthy

68.9k53169

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hmm… the definitions I’ve seen only required $X$ map to a measurable space $(A, mathcal A)$, not necessarily $A = mathbb R$ and not necessarily $mathcal A = mathcal B (mathbb R)$.
  $endgroup$
  – Yatharth Agarwal
  Mar 13 at 8:07
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you provide a reference? I’d love to dig deeper.
  $endgroup$
  – Yatharth Agarwal
  Mar 13 at 8:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  In that case you are asking about existence of what? Can you give a precise definition of the term 'random variable' and the term 'distribution function'?
  $endgroup$
  – Kavi Rama Murthy
  Mar 13 at 8:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @YatharthAgarwal What you have defined is often called a random element of $A$. In any case please see if my revised answer is clear.
  $endgroup$
  – Kavi Rama Murthy
  Mar 13 at 8:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ahhh random elements vs random variables. The text I was using didn’t distinguish. This makes sense—thank you!
  $endgroup$
  – Yatharth Agarwal
  Mar 13 at 8:19
  

  
  
  
  

add a comment | 

$begingroup$

What you have defined is often called a random element of $A$. In contrast, random variables are usually defined as measurable functions from a probability space to $(mathbb R, mathcal B(mathbb R))$. With this restriction, the definition $F_X(x)=P(Xleq x)$ is always valid.

If a random variable is defined as measurable map from a measure space into a measurable space the we can associate with it what is called the induced measure: $nu (A)=P(X^{-1}(A))$ for $A in mathcal A$. Sometimes this is also called the distribution of $X$. It so happens that in the case $A=mathbb R$ with Borel sigma algebra there is a one-to-one correspondence between induced measures and the functions $F_X$.

share|cite|improve this answer

edited Mar 13 at 8:22

Yatharth Agarwal

542418

answered Mar 13 at 8:04

Kavi Rama MurthyKavi Rama Murthy

68.9k53169

$endgroup$

share|cite|improve this answer

edited Mar 13 at 8:22

Yatharth Agarwal

542418

answered Mar 13 at 8:04

Kavi Rama MurthyKavi Rama Murthy

68.9k53169

edited Mar 13 at 8:22

Yatharth Agarwal

542418

edited Mar 13 at 8:22

Yatharth Agarwal

542418

answered Mar 13 at 8:04

Kavi Rama MurthyKavi Rama Murthy

68.9k53169

answered Mar 13 at 8:04

Kavi Rama MurthyKavi Rama Murthy

68.9k53169