Proof that when $n to infty$, then $P(A_{n}cap B_{n})to p$
$begingroup$
Show that $A_1$, $A_2$, ... and $B_1$, $B_2$ ... are all aleatory events of the same probability space such that $P(A_{n}) to 1$ and $P (B_{n})to p$, when $n to infty$ then $P(A_{n}cap B_{n})to p$.
I know there is a theorem that says:
If the sequence $(A_{n})_{n>1}$, where $A_{n} in mathbb{A}$, decrease to the empty set, $A_{n} supset A_{n + 1}$ for all n, then $P (A_{n}) to 0$ when $n to infty$.
I am almost sure that I have to use this Theorem but I am not able to do this.
Any help?
probability-theory elementary-set-theory convergence
edited Jan 8 at 21:09
Davide Giraudo
127k16151264
asked Jan 8 at 19:46
LauraLaura
2578
$endgroup$
add a comment |
$begingroup$
Show that $A_1$, $A_2$, ... and $B_1$, $B_2$ ... are all aleatory events of the same probability space such that $P(A_{n}) to 1$ and $P (B_{n})to p$, when $n to infty$ then $P(A_{n}cap B_{n})to p$.
I know there is a theorem that says:
If the sequence $(A_{n})_{n>1}$, where $A_{n} in mathbb{A}$, decrease to the empty set, $A_{n} supset A_{n + 1}$ for all n, then $P (A_{n}) to 0$ when $n to infty$.
I am almost sure that I have to use this Theorem but I am not able to do this.
Any help?
probability-theory elementary-set-theory convergence
edited Jan 8 at 21:09
Davide Giraudo
127k16151264
asked Jan 8 at 19:46
LauraLaura
2578
$endgroup$
add a comment |
$begingroup$
Show that $A_1$, $A_2$, ... and $B_1$, $B_2$ ... are all aleatory events of the same probability space such that $P(A_{n}) to 1$ and $P (B_{n})to p$, when $n to infty$ then $P(A_{n}cap B_{n})to p$.
I know there is a theorem that says:
If the sequence $(A_{n})_{n>1}$, where $A_{n} in mathbb{A}$, decrease to the empty set, $A_{n} supset A_{n + 1}$ for all n, then $P (A_{n}) to 0$ when $n to infty$.
I am almost sure that I have to use this Theorem but I am not able to do this.
Any help?
probability-theory elementary-set-theory convergence
edited Jan 8 at 21:09
Davide Giraudo
127k16151264
asked Jan 8 at 19:46
LauraLaura
2578
$endgroup$
Show that $A_1$, $A_2$, ... and $B_1$, $B_2$ ... are all aleatory events of the same probability space such that $P(A_{n}) to 1$ and $P (B_{n})to p$, when $n to infty$ then $P(A_{n}cap B_{n})to p$.
I know there is a theorem that says:
If the sequence $(A_{n})_{n>1}$, where $A_{n} in mathbb{A}$, decrease to the empty set, $A_{n} supset A_{n + 1}$ for all n, then $P (A_{n}) to 0$ when $n to infty$.
I am almost sure that I have to use this Theorem but I am not able to do this.
Any help?
probability-theory elementary-set-theory convergence
probability-theory elementary-set-theory convergence
edited Jan 8 at 21:09
Davide Giraudo
127k16151264
asked Jan 8 at 19:46
LauraLaura
2578
edited Jan 8 at 21:09
Davide Giraudo
127k16151264
asked Jan 8 at 19:46
LauraLaura
2578
edited Jan 8 at 21:09
Davide Giraudo
127k16151264
edited Jan 8 at 21:09
Davide Giraudo
127k16151264
edited Jan 8 at 21:09
Davide Giraudo
127k16151264
127k16151264
asked Jan 8 at 19:46
LauraLaura
2578
asked Jan 8 at 19:46
LauraLaura
2578
asked Jan 8 at 19:46
LauraLaura
2578
2578
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Unfortunately there is no assumption of non-increasingness /decreasingness hence we cannot use directly this result. However, notice that
$$
mathbb Pleft(A_ncap B_nright)+mathbb Pleft(A_n^ccap B_nright)=mathbb Pleft(B_nright)
$$
hence
$$
mathbb Pleft(A_ncap B_nright)=mathbb Pleft(B_nright)-mathbb Pleft(A_n^ccap B_nright).
$$
The only remaining question is: what is $lim_{nto +infty}mathbb Pleft(A_n^ccap B_nright)$?
answered Jan 8 at 19:57
Davide GiraudoDavide Giraudo
127k16151264
$endgroup$
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
});
}
});
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%2f3066635%2fproof-that-when-n-to-infty-then-pa-n-cap-b-n-to-p%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
$begingroup$
Unfortunately there is no assumption of non-increasingness /decreasingness hence we cannot use directly this result. However, notice that
$$
mathbb Pleft(A_ncap B_nright)+mathbb Pleft(A_n^ccap B_nright)=mathbb Pleft(B_nright)
$$
hence
$$
mathbb Pleft(A_ncap B_nright)=mathbb Pleft(B_nright)-mathbb Pleft(A_n^ccap B_nright).
$$
The only remaining question is: what is $lim_{nto +infty}mathbb Pleft(A_n^ccap B_nright)$?
answered Jan 8 at 19:57
Davide GiraudoDavide Giraudo
127k16151264
$endgroup$
add a comment |
$begingroup$
Unfortunately there is no assumption of non-increasingness /decreasingness hence we cannot use directly this result. However, notice that
$$
mathbb Pleft(A_ncap B_nright)+mathbb Pleft(A_n^ccap B_nright)=mathbb Pleft(B_nright)
$$
hence
$$
mathbb Pleft(A_ncap B_nright)=mathbb Pleft(B_nright)-mathbb Pleft(A_n^ccap B_nright).
$$
The only remaining question is: what is $lim_{nto +infty}mathbb Pleft(A_n^ccap B_nright)$?
answered Jan 8 at 19:57
Davide GiraudoDavide Giraudo
127k16151264
$endgroup$
add a comment |
$begingroup$
Unfortunately there is no assumption of non-increasingness /decreasingness hence we cannot use directly this result. However, notice that
$$
mathbb Pleft(A_ncap B_nright)+mathbb Pleft(A_n^ccap B_nright)=mathbb Pleft(B_nright)
$$
hence
$$
mathbb Pleft(A_ncap B_nright)=mathbb Pleft(B_nright)-mathbb Pleft(A_n^ccap B_nright).
$$
The only remaining question is: what is $lim_{nto +infty}mathbb Pleft(A_n^ccap B_nright)$?
answered Jan 8 at 19:57
Davide GiraudoDavide Giraudo
127k16151264
$endgroup$
Unfortunately there is no assumption of non-increasingness /decreasingness hence we cannot use directly this result. However, notice that
$$
mathbb Pleft(A_ncap B_nright)+mathbb Pleft(A_n^ccap B_nright)=mathbb Pleft(B_nright)
$$
hence
$$
mathbb Pleft(A_ncap B_nright)=mathbb Pleft(B_nright)-mathbb Pleft(A_n^ccap B_nright).
$$
The only remaining question is: what is $lim_{nto +infty}mathbb Pleft(A_n^ccap B_nright)$?
answered Jan 8 at 19:57
Davide GiraudoDavide Giraudo
127k16151264
answered Jan 8 at 19:57
Davide GiraudoDavide Giraudo
127k16151264
answered Jan 8 at 19:57
Davide GiraudoDavide Giraudo
127k16151264
answered Jan 8 at 19:57
Davide GiraudoDavide Giraudo
127k16151264
127k16151264
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.
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%2f3066635%2fproof-that-when-n-to-infty-then-pa-n-cap-b-n-to-p%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
