Understanding a $sigma$-field
Problem
Let's consider $n$ Bernoulli trials with the probability of getting a success equal to $p$. My task is to find the expected value of getting a success in the first trial on condition that we know how many successes have occurred in all trials.
My attempt
The problem is quite easy. I define two variables
$S_n$ - the amount of successes,
$X$ - the amount of successes in the first trial.
I would like to find the value of $mathbb{E}(X | sigma(S_n))$.
Now I would like to do something with $sigma(S_n)$. I defined new sequence of events:
$$A_k = {S_n = k} tag{1}.$$
Now $sigma(S_n) = sigma(A_0, ldots, A_n)$.
The further calculations are very easy. The answer to this problem is $frac{S_n}{n}$.
What I don't understand
I don't really understand how does $sigma(A_0, ldots, A_n)$ look like. To my mind there are a lot of events that are completely impossible. Let me give you an example.
Let's fix $n=1$. That gives us:
$$A_0, A_1$$
and
$$sigma(A_0, A_1) = {emptyset, Omega = A_0 cup A_1, A_0, A_1 }.$$
Am I correct? How can I interpret $A_0 cup A_1$?
probability probability-theory conditional-expectation
edited Nov 28 at 6:54
Shashi
7,0361528
asked Nov 26 at 17:08
Hendrra
1,079416
add a comment |
Problem
Let's consider $n$ Bernoulli trials with the probability of getting a success equal to $p$. My task is to find the expected value of getting a success in the first trial on condition that we know how many successes have occurred in all trials.
My attempt
The problem is quite easy. I define two variables
$S_n$ - the amount of successes,
$X$ - the amount of successes in the first trial.
I would like to find the value of $mathbb{E}(X | sigma(S_n))$.
Now I would like to do something with $sigma(S_n)$. I defined new sequence of events:
$$A_k = {S_n = k} tag{1}.$$
Now $sigma(S_n) = sigma(A_0, ldots, A_n)$.
The further calculations are very easy. The answer to this problem is $frac{S_n}{n}$.
What I don't understand
I don't really understand how does $sigma(A_0, ldots, A_n)$ look like. To my mind there are a lot of events that are completely impossible. Let me give you an example.
Let's fix $n=1$. That gives us:
$$A_0, A_1$$
and
$$sigma(A_0, A_1) = {emptyset, Omega = A_0 cup A_1, A_0, A_1 }.$$
Am I correct? How can I interpret $A_0 cup A_1$?
probability probability-theory conditional-expectation
edited Nov 28 at 6:54
Shashi
7,0361528
asked Nov 26 at 17:08
Hendrra
1,079416
add a comment |
Problem
Let's consider $n$ Bernoulli trials with the probability of getting a success equal to $p$. My task is to find the expected value of getting a success in the first trial on condition that we know how many successes have occurred in all trials.
My attempt
The problem is quite easy. I define two variables
$S_n$ - the amount of successes,
$X$ - the amount of successes in the first trial.
I would like to find the value of $mathbb{E}(X | sigma(S_n))$.
Now I would like to do something with $sigma(S_n)$. I defined new sequence of events:
$$A_k = {S_n = k} tag{1}.$$
Now $sigma(S_n) = sigma(A_0, ldots, A_n)$.
The further calculations are very easy. The answer to this problem is $frac{S_n}{n}$.
What I don't understand
I don't really understand how does $sigma(A_0, ldots, A_n)$ look like. To my mind there are a lot of events that are completely impossible. Let me give you an example.
Let's fix $n=1$. That gives us:
$$A_0, A_1$$
and
$$sigma(A_0, A_1) = {emptyset, Omega = A_0 cup A_1, A_0, A_1 }.$$
Am I correct? How can I interpret $A_0 cup A_1$?
probability probability-theory conditional-expectation
edited Nov 28 at 6:54
Shashi
7,0361528
asked Nov 26 at 17:08
Hendrra
1,079416
Problem
Let's consider $n$ Bernoulli trials with the probability of getting a success equal to $p$. My task is to find the expected value of getting a success in the first trial on condition that we know how many successes have occurred in all trials.
My attempt
The problem is quite easy. I define two variables
$S_n$ - the amount of successes,
$X$ - the amount of successes in the first trial.
I would like to find the value of $mathbb{E}(X | sigma(S_n))$.
Now I would like to do something with $sigma(S_n)$. I defined new sequence of events:
$$A_k = {S_n = k} tag{1}.$$
Now $sigma(S_n) = sigma(A_0, ldots, A_n)$.
The further calculations are very easy. The answer to this problem is $frac{S_n}{n}$.
What I don't understand
I don't really understand how does $sigma(A_0, ldots, A_n)$ look like. To my mind there are a lot of events that are completely impossible. Let me give you an example.
Let's fix $n=1$. That gives us:
$$A_0, A_1$$
and
$$sigma(A_0, A_1) = {emptyset, Omega = A_0 cup A_1, A_0, A_1 }.$$
Am I correct? How can I interpret $A_0 cup A_1$?
probability probability-theory conditional-expectation
probability probability-theory conditional-expectation
edited Nov 28 at 6:54
Shashi
7,0361528
asked Nov 26 at 17:08
Hendrra
1,079416
edited Nov 28 at 6:54
Shashi
7,0361528
asked Nov 26 at 17:08
Hendrra
1,079416
edited Nov 28 at 6:54
Shashi
7,0361528
edited Nov 28 at 6:54
Shashi
7,0361528
edited Nov 28 at 6:54
Shashi
7,0361528
7,0361528
asked Nov 26 at 17:08
Hendrra
1,079416
asked Nov 26 at 17:08
Hendrra
1,079416
asked Nov 26 at 17:08
Hendrra
1,079416
1,079416
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
If $S_1$ can't take any values other than $0$ and $1$ (even on null sets) then you are correct.
$A_0 cup A_1$ is the event that either $S_1 = 0$ or $S_1 = 1$ occurs (or both, which in this case isn't possible). You can always think of unions as "or" and of intersections as "and".
In general, $sigma(S_n)$ could be interpreted as the smallest $sigma$-algebra that contains all the information about $S_n$.
answered Nov 26 at 20:01
Tki Deneb
26710
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
});
}
});
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%2f3014605%2funderstanding-a-sigma-field%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
If $S_1$ can't take any values other than $0$ and $1$ (even on null sets) then you are correct.
$A_0 cup A_1$ is the event that either $S_1 = 0$ or $S_1 = 1$ occurs (or both, which in this case isn't possible). You can always think of unions as "or" and of intersections as "and".
In general, $sigma(S_n)$ could be interpreted as the smallest $sigma$-algebra that contains all the information about $S_n$.
answered Nov 26 at 20:01
Tki Deneb
26710
add a comment |
If $S_1$ can't take any values other than $0$ and $1$ (even on null sets) then you are correct.
$A_0 cup A_1$ is the event that either $S_1 = 0$ or $S_1 = 1$ occurs (or both, which in this case isn't possible). You can always think of unions as "or" and of intersections as "and".
In general, $sigma(S_n)$ could be interpreted as the smallest $sigma$-algebra that contains all the information about $S_n$.
answered Nov 26 at 20:01
Tki Deneb
26710
add a comment |
If $S_1$ can't take any values other than $0$ and $1$ (even on null sets) then you are correct.
$A_0 cup A_1$ is the event that either $S_1 = 0$ or $S_1 = 1$ occurs (or both, which in this case isn't possible). You can always think of unions as "or" and of intersections as "and".
In general, $sigma(S_n)$ could be interpreted as the smallest $sigma$-algebra that contains all the information about $S_n$.
answered Nov 26 at 20:01
Tki Deneb
26710
If $S_1$ can't take any values other than $0$ and $1$ (even on null sets) then you are correct.
$A_0 cup A_1$ is the event that either $S_1 = 0$ or $S_1 = 1$ occurs (or both, which in this case isn't possible). You can always think of unions as "or" and of intersections as "and".
In general, $sigma(S_n)$ could be interpreted as the smallest $sigma$-algebra that contains all the information about $S_n$.
answered Nov 26 at 20:01
Tki Deneb
26710
edited Nov 27 at 18:25
answered Nov 26 at 20:01
Tki Deneb
26710
answered Nov 26 at 20:01
Tki Deneb
26710
answered Nov 26 at 20:01
Tki Deneb
26710
26710
add a comment |
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3014605%2funderstanding-a-sigma-field%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
