Mixing
How is the random variable $X=max{x_1,dots,x_n}$ related to the negative binomial distribution
$begingroup$
Consider a jar with $N$ indexed balls from $1$ to $N$ and the variable $X=max{x_1,dots,x_n} $ where $n<N$ is the number of balls we took out without replacement.
I found that $P(X=k)=begin{cases} 0 &text{ if } k<n\ frac{{k-1choose n-1}}{Nchoose n-1} &text { if } nle kle N end{cases}$
Is $X$ related to the negative binomial distribution, and how?
probability negative-binomial
asked Jan 26 at 13:33
H. WalterH. Walter
1047
$endgroup$
add a comment |
$begingroup$
Consider a jar with $N$ indexed balls from $1$ to $N$ and the variable $X=max{x_1,dots,x_n} $ where $n<N$ is the number of balls we took out without replacement.
I found that $P(X=k)=begin{cases} 0 &text{ if } k<n\ frac{{k-1choose n-1}}{Nchoose n-1} &text { if } nle kle N end{cases}$
Is $X$ related to the negative binomial distribution, and how?
probability negative-binomial
asked Jan 26 at 13:33
H. WalterH. Walter
1047
$endgroup$
$begingroup$
What are the $x_i$?
$endgroup$
– David M.
Jan 26 at 20:55
$begingroup$
the balls we took out
$endgroup$
– H. Walter
Jan 27 at 7:20
$begingroup$
Ah okay. I don’t see a connection with negative binomial, as the trials in a negative-binomial-type experiment are independent. The sampling without replacement makes the trials dependent (e.g. if you sample ball $N$ on the first go, $X$ becomes degenerate).
$endgroup$
– David M.
Jan 27 at 14:33
$begingroup$
Since the sampling is without replacement, perhaps you should consider the relation to HyperGeometric distribution.
$endgroup$
– Lee David Chung Lin
Jan 29 at 9:28
add a comment |
$begingroup$
Consider a jar with $N$ indexed balls from $1$ to $N$ and the variable $X=max{x_1,dots,x_n} $ where $n<N$ is the number of balls we took out without replacement.
I found that $P(X=k)=begin{cases} 0 &text{ if } k<n\ frac{{k-1choose n-1}}{Nchoose n-1} &text { if } nle kle N end{cases}$
Is $X$ related to the negative binomial distribution, and how?
probability negative-binomial
asked Jan 26 at 13:33
H. WalterH. Walter
1047
$endgroup$
Consider a jar with $N$ indexed balls from $1$ to $N$ and the variable $X=max{x_1,dots,x_n} $ where $n<N$ is the number of balls we took out without replacement.
I found that $P(X=k)=begin{cases} 0 &text{ if } k<n\ frac{{k-1choose n-1}}{Nchoose n-1} &text { if } nle kle N end{cases}$
Is $X$ related to the negative binomial distribution, and how?
probability negative-binomial
probability negative-binomial
asked Jan 26 at 13:33
H. WalterH. Walter
1047
asked Jan 26 at 13:33
H. WalterH. Walter
1047
asked Jan 26 at 13:33
H. WalterH. Walter
1047
asked Jan 26 at 13:33
H. WalterH. Walter
1047
asked Jan 26 at 13:33
H. WalterH. Walter
1047
1047
$begingroup$
What are the $x_i$?
$endgroup$
– David M.
Jan 26 at 20:55
$begingroup$
the balls we took out
$endgroup$
– H. Walter
Jan 27 at 7:20
$begingroup$
Ah okay. I don’t see a connection with negative binomial, as the trials in a negative-binomial-type experiment are independent. The sampling without replacement makes the trials dependent (e.g. if you sample ball $N$ on the first go, $X$ becomes degenerate).
$endgroup$
– David M.
Jan 27 at 14:33
$begingroup$
Since the sampling is without replacement, perhaps you should consider the relation to HyperGeometric distribution.
$endgroup$
– Lee David Chung Lin
Jan 29 at 9:28
add a comment |
$begingroup$
What are the $x_i$?
$endgroup$
– David M.
Jan 26 at 20:55
$begingroup$
the balls we took out
$endgroup$
– H. Walter
Jan 27 at 7:20
$begingroup$
Ah okay. I don’t see a connection with negative binomial, as the trials in a negative-binomial-type experiment are independent. The sampling without replacement makes the trials dependent (e.g. if you sample ball $N$ on the first go, $X$ becomes degenerate).
$endgroup$
– David M.
Jan 27 at 14:33
$begingroup$
Since the sampling is without replacement, perhaps you should consider the relation to HyperGeometric distribution.
$endgroup$
– Lee David Chung Lin
Jan 29 at 9:28
$begingroup$
What are the $x_i$?
$endgroup$
– David M.
Jan 26 at 20:55
$begingroup$
What are the $x_i$?
$endgroup$
– David M.
Jan 26 at 20:55
$begingroup$
the balls we took out
$endgroup$
– H. Walter
Jan 27 at 7:20
$begingroup$
the balls we took out
$endgroup$
– H. Walter
Jan 27 at 7:20
$begingroup$
Ah okay. I don’t see a connection with negative binomial, as the trials in a negative-binomial-type experiment are independent. The sampling without replacement makes the trials dependent (e.g. if you sample ball $N$ on the first go, $X$ becomes degenerate).
$endgroup$
– David M.
Jan 27 at 14:33
$begingroup$
Ah okay. I don’t see a connection with negative binomial, as the trials in a negative-binomial-type experiment are independent. The sampling without replacement makes the trials dependent (e.g. if you sample ball $N$ on the first go, $X$ becomes degenerate).
$endgroup$
– David M.
Jan 27 at 14:33
$begingroup$
Since the sampling is without replacement, perhaps you should consider the relation to HyperGeometric distribution.
$endgroup$
– Lee David Chung Lin
Jan 29 at 9:28
$begingroup$
Since the sampling is without replacement, perhaps you should consider the relation to HyperGeometric distribution.
$endgroup$
– Lee David Chung Lin
Jan 29 at 9:28
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%2f3088248%2fhow-is-the-random-variable-x-max-x-1-dots-x-n-related-to-the-negative-bin%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%2f3088248%2fhow-is-the-random-variable-x-max-x-1-dots-x-n-related-to-the-negative-bin%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
$begingroup$
What are the $x_i$?
$endgroup$
– David M.
Jan 26 at 20:55
$begingroup$
the balls we took out
$endgroup$
– H. Walter
Jan 27 at 7:20
$begingroup$
Ah okay. I don’t see a connection with negative binomial, as the trials in a negative-binomial-type experiment are independent. The sampling without replacement makes the trials dependent (e.g. if you sample ball $N$ on the first go, $X$ becomes degenerate).
$endgroup$
– David M.
Jan 27 at 14:33
$begingroup$
Since the sampling is without replacement, perhaps you should consider the relation to HyperGeometric distribution.
$endgroup$
– Lee David Chung Lin
Jan 29 at 9:28