Proof of a lower bound on probability
$begingroup$
How to prove that the probability of simultaneous occurrence of more than $frac n 2$ events from $n$ independent Bernoulli trials is greater than or equal to:$$1-e^{-2nleft(p-frac{1}{2}right)^2}$$ where $p$ is the probability of occurrence of each Bernoulli trial.
PS: I tried proving this using the Chernoff bound but without any success. I could prove the lower bound to be $$1-e^{-frac{1}{2p}nleft(p-frac{1}{2}right)^2}$$. My approach is similar to the answer provided here.
probability probability-distributions upper-lower-bounds
asked Jan 20 at 9:31
Durgesh AgrawalDurgesh Agrawal
22
$endgroup$
add a comment |
$begingroup$
How to prove that the probability of simultaneous occurrence of more than $frac n 2$ events from $n$ independent Bernoulli trials is greater than or equal to:$$1-e^{-2nleft(p-frac{1}{2}right)^2}$$ where $p$ is the probability of occurrence of each Bernoulli trial.
PS: I tried proving this using the Chernoff bound but without any success. I could prove the lower bound to be $$1-e^{-frac{1}{2p}nleft(p-frac{1}{2}right)^2}$$. My approach is similar to the answer provided here.
probability probability-distributions upper-lower-bounds
asked Jan 20 at 9:31
Durgesh AgrawalDurgesh Agrawal
22
$endgroup$
$begingroup$
Can you show your working for what you've done please?
$endgroup$
– stuart stevenson
Jan 20 at 10:28
$begingroup$
@JoséCarlosSantos I hope the question is complete now.
$endgroup$
– Durgesh Agrawal
Jan 20 at 10:40
add a comment |
$begingroup$
How to prove that the probability of simultaneous occurrence of more than $frac n 2$ events from $n$ independent Bernoulli trials is greater than or equal to:$$1-e^{-2nleft(p-frac{1}{2}right)^2}$$ where $p$ is the probability of occurrence of each Bernoulli trial.
PS: I tried proving this using the Chernoff bound but without any success. I could prove the lower bound to be $$1-e^{-frac{1}{2p}nleft(p-frac{1}{2}right)^2}$$. My approach is similar to the answer provided here.
probability probability-distributions upper-lower-bounds
asked Jan 20 at 9:31
Durgesh AgrawalDurgesh Agrawal
22
$endgroup$
How to prove that the probability of simultaneous occurrence of more than $frac n 2$ events from $n$ independent Bernoulli trials is greater than or equal to:$$1-e^{-2nleft(p-frac{1}{2}right)^2}$$ where $p$ is the probability of occurrence of each Bernoulli trial.
PS: I tried proving this using the Chernoff bound but without any success. I could prove the lower bound to be $$1-e^{-frac{1}{2p}nleft(p-frac{1}{2}right)^2}$$. My approach is similar to the answer provided here.
probability probability-distributions upper-lower-bounds
probability probability-distributions upper-lower-bounds
asked Jan 20 at 9:31
Durgesh AgrawalDurgesh Agrawal
22
asked Jan 20 at 9:31
Durgesh AgrawalDurgesh Agrawal
22
edited Jan 20 at 10:34
Durgesh Agrawal
asked Jan 20 at 9:31
Durgesh AgrawalDurgesh Agrawal
22
asked Jan 20 at 9:31
Durgesh AgrawalDurgesh Agrawal
22
asked Jan 20 at 9:31
Durgesh AgrawalDurgesh Agrawal
22
22
$begingroup$
Can you show your working for what you've done please?
$endgroup$
– stuart stevenson
Jan 20 at 10:28
$begingroup$
@JoséCarlosSantos I hope the question is complete now.
$endgroup$
– Durgesh Agrawal
Jan 20 at 10:40
add a comment |
$begingroup$
Can you show your working for what you've done please?
$endgroup$
– stuart stevenson
Jan 20 at 10:28
$begingroup$
@JoséCarlosSantos I hope the question is complete now.
$endgroup$
– Durgesh Agrawal
Jan 20 at 10:40
$begingroup$
Can you show your working for what you've done please?
$endgroup$
– stuart stevenson
Jan 20 at 10:28
$begingroup$
Can you show your working for what you've done please?
$endgroup$
– stuart stevenson
Jan 20 at 10:28
$begingroup$
@JoséCarlosSantos I hope the question is complete now.
$endgroup$
– Durgesh Agrawal
Jan 20 at 10:40
$begingroup$
@JoséCarlosSantos I hope the question is complete now.
$endgroup$
– Durgesh Agrawal
Jan 20 at 10:40
add a comment |
0
active
oldest
votes
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%2f3080366%2fproof-of-a-lower-bound-on-probability%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3080366%2fproof-of-a-lower-bound-on-probability%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
$begingroup$
Can you show your working for what you've done please?
$endgroup$
– stuart stevenson
Jan 20 at 10:28
$begingroup$
@JoséCarlosSantos I hope the question is complete now.
$endgroup$
– Durgesh Agrawal
Jan 20 at 10:40