$k(ln (k) - 1) = ln (2n)$ The Next CEO of Stack OverflowA multinomial problem (balls, bins, etc.)Calculating the distribution of the minimum of two exponential functionsGiven $f_Y(y;lambda) = lambda e^-lambda y$ , $y > 0 $, show $hat lambda = Y_1$ is not consistent for $lambda$Derivation of Likelihood Function for Random Effects ParametersFrog jumping between states, expected timeThe step of swaping terms in the Rademacher boundConditional convergence of $Bbb E[X^-1]$ for $XsimmathcalN(mu,sigma^2)$ as $operatornamepr(X>0)to 1$What is what between binomial many flip outcomes and statistical observation?Is explosive AR(1) stationary?How to interpret the proof that information cascades will form?
How exploitable/balanced is this homebrew spell: Spell Permanency?
Is there a rule of thumb for determining the amount one should accept for of a settlement offer?
Is it possible to make a 9x9 table fit within the default margins?
Why doesn't Shulchan Aruch include the laws of destroying fruit trees?
Masking layers by a vector polygon layer in QGIS
Does Germany produce more waste than the US?
Cannot restore registry to default in Windows 10?
"Eavesdropping" vs "Listen in on"
Create custom note boxes
pgfplots: How to draw a tangent graph below two others?
Upgrading From a 9 Speed Sora Derailleur?
MT "will strike" & LXX "will watch carefully" (Gen 3:15)?
Does the Idaho Potato Commission associate potato skins with healthy eating?
What day is it again?
Can a PhD from a non-TU9 German university become a professor in a TU9 university?
Traveling with my 5 year old daughter (as the father) without the mother from Germany to Mexico
How can a day be of 24 hours?
How should I connect my cat5 cable to connectors having an orange-green line?
Avoiding the "not like other girls" trope?
Why was Sir Cadogan fired?
Man transported from Alternate World into ours by a Neutrino Detector
Can you teleport closer to a creature you are Frightened of?
Compilation of a 2d array and a 1d array
Creating a script with console commands
$k(ln (k) - 1) = ln (2n)$
The Next CEO of Stack OverflowA multinomial problem (balls, bins, etc.)Calculating the distribution of the minimum of two exponential functionsGiven $f_Y(y;lambda) = lambda e^-lambda y$ , $y > 0 $, show $hat lambda = Y_1$ is not consistent for $lambda$Derivation of Likelihood Function for Random Effects ParametersFrog jumping between states, expected timeThe step of swaping terms in the Rademacher boundConditional convergence of $Bbb E[X^-1]$ for $XsimmathcalN(mu,sigma^2)$ as $operatornamepr(X>0)to 1$What is what between binomial many flip outcomes and statistical observation?Is explosive AR(1) stationary?How to interpret the proof that information cascades will form?
$begingroup$
On the last page of
https://www.eecs70.org/static/notes/n17.pdf
I don't understand where the following claim comes from.
$ln k! ≈ ln 2n$ if $k$ is chosen to be $ln n / ln ln n$
Can someone prove it / explain why $ln n/ln ln n$ is a reasonable $k$ value?
Basically, why $ln n/ ln ln n$ is a good choice for $k$ in the equality in the title? (The LHS is after applying Stirling's approx to $k!$)
probability
asked Mar 20 at 7:31
ajfbiw.sajfbiw.s
857
$endgroup$
add a comment |
$begingroup$
On the last page of
https://www.eecs70.org/static/notes/n17.pdf
I don't understand where the following claim comes from.
$ln k! ≈ ln 2n$ if $k$ is chosen to be $ln n / ln ln n$
Can someone prove it / explain why $ln n/ln ln n$ is a reasonable $k$ value?
Basically, why $ln n/ ln ln n$ is a good choice for $k$ in the equality in the title? (The LHS is after applying Stirling's approx to $k!$)
probability
asked Mar 20 at 7:31
ajfbiw.sajfbiw.s
857
$endgroup$
add a comment |
$begingroup$
On the last page of
https://www.eecs70.org/static/notes/n17.pdf
I don't understand where the following claim comes from.
$ln k! ≈ ln 2n$ if $k$ is chosen to be $ln n / ln ln n$
Can someone prove it / explain why $ln n/ln ln n$ is a reasonable $k$ value?
Basically, why $ln n/ ln ln n$ is a good choice for $k$ in the equality in the title? (The LHS is after applying Stirling's approx to $k!$)
probability
asked Mar 20 at 7:31
ajfbiw.sajfbiw.s
857
$endgroup$
On the last page of
https://www.eecs70.org/static/notes/n17.pdf
I don't understand where the following claim comes from.
$ln k! ≈ ln 2n$ if $k$ is chosen to be $ln n / ln ln n$
Can someone prove it / explain why $ln n/ln ln n$ is a reasonable $k$ value?
Basically, why $ln n/ ln ln n$ is a good choice for $k$ in the equality in the title? (The LHS is after applying Stirling's approx to $k!$)
probability
probability
asked Mar 20 at 7:31
ajfbiw.sajfbiw.s
857
asked Mar 20 at 7:31
ajfbiw.sajfbiw.s
857
edited Mar 20 at 19:07
ajfbiw.s
asked Mar 20 at 7:31
ajfbiw.sajfbiw.s
857
asked Mar 20 at 7:31
ajfbiw.sajfbiw.s
857
asked Mar 20 at 7:31
ajfbiw.sajfbiw.s
857
857
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
By Stirling,
$$ln k!approx k(ln k-1).$$
Then
$$ln k!=lnfracln nlnln n!approxfracln nlnln nleft(lnfracln nlnln n-1right)=ln nfraclnln n-lnlnln n-1lnln napprox ln n$$
I have no convincing explanation for the factor $2$. I have the feeling that $k$ should have been chosen as
$$fracln 2nlnln 2n$$ though in practice this makes little difference.
Numerical example:
Let $n=10^9$.
We can take
$$k=fracln 10^9lnln10^9=6.837cdots$$
or
$$k=fracln 2cdot10^9lnln2cdot10^9=6.989cdots$$
Then in both cases, $7!=5040$ is a pretty poor approximation.
answered Mar 20 at 8:05
Yves DaoustYves Daoust
131k676229
$endgroup$
add a comment |
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%2f3155127%2fk-ln-k-1-ln-2n%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
$begingroup$
By Stirling,
$$ln k!approx k(ln k-1).$$
Then
$$ln k!=lnfracln nlnln n!approxfracln nlnln nleft(lnfracln nlnln n-1right)=ln nfraclnln n-lnlnln n-1lnln napprox ln n$$
I have no convincing explanation for the factor $2$. I have the feeling that $k$ should have been chosen as
$$fracln 2nlnln 2n$$ though in practice this makes little difference.
Numerical example:
Let $n=10^9$.
We can take
$$k=fracln 10^9lnln10^9=6.837cdots$$
or
$$k=fracln 2cdot10^9lnln2cdot10^9=6.989cdots$$
Then in both cases, $7!=5040$ is a pretty poor approximation.
answered Mar 20 at 8:05
Yves DaoustYves Daoust
131k676229
$endgroup$
add a comment |
$begingroup$
By Stirling,
$$ln k!approx k(ln k-1).$$
Then
$$ln k!=lnfracln nlnln n!approxfracln nlnln nleft(lnfracln nlnln n-1right)=ln nfraclnln n-lnlnln n-1lnln napprox ln n$$
I have no convincing explanation for the factor $2$. I have the feeling that $k$ should have been chosen as
$$fracln 2nlnln 2n$$ though in practice this makes little difference.
Numerical example:
Let $n=10^9$.
We can take
$$k=fracln 10^9lnln10^9=6.837cdots$$
or
$$k=fracln 2cdot10^9lnln2cdot10^9=6.989cdots$$
Then in both cases, $7!=5040$ is a pretty poor approximation.
answered Mar 20 at 8:05
Yves DaoustYves Daoust
131k676229
$endgroup$
add a comment |
$begingroup$
By Stirling,
$$ln k!approx k(ln k-1).$$
Then
$$ln k!=lnfracln nlnln n!approxfracln nlnln nleft(lnfracln nlnln n-1right)=ln nfraclnln n-lnlnln n-1lnln napprox ln n$$
I have no convincing explanation for the factor $2$. I have the feeling that $k$ should have been chosen as
$$fracln 2nlnln 2n$$ though in practice this makes little difference.
Numerical example:
Let $n=10^9$.
We can take
$$k=fracln 10^9lnln10^9=6.837cdots$$
or
$$k=fracln 2cdot10^9lnln2cdot10^9=6.989cdots$$
Then in both cases, $7!=5040$ is a pretty poor approximation.
answered Mar 20 at 8:05
Yves DaoustYves Daoust
131k676229
$endgroup$
By Stirling,
$$ln k!approx k(ln k-1).$$
Then
$$ln k!=lnfracln nlnln n!approxfracln nlnln nleft(lnfracln nlnln n-1right)=ln nfraclnln n-lnlnln n-1lnln napprox ln n$$
I have no convincing explanation for the factor $2$. I have the feeling that $k$ should have been chosen as
$$fracln 2nlnln 2n$$ though in practice this makes little difference.
Numerical example:
Let $n=10^9$.
We can take
$$k=fracln 10^9lnln10^9=6.837cdots$$
or
$$k=fracln 2cdot10^9lnln2cdot10^9=6.989cdots$$
Then in both cases, $7!=5040$ is a pretty poor approximation.
answered Mar 20 at 8:05
Yves DaoustYves Daoust
131k676229
edited Mar 20 at 8:20
answered Mar 20 at 8:05
Yves DaoustYves Daoust
131k676229
answered Mar 20 at 8:05
Yves DaoustYves Daoust
131k676229
answered Mar 20 at 8:05
Yves DaoustYves Daoust
131k676229
131k676229
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%2f3155127%2fk-ln-k-1-ln-2n%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
