Mixing
Finding marginal density from probability generating function of two random variables X and Y
0
$begingroup$
Given the probability generating function of two random variables $(X,Y)$ as $G_{X,Y} (u,v)$, how to find the marginal densities of X and Y?
Specifying the probability generating function:
$G_{X,Y} (u,v) = e^{-lambda_{1} -lambda_{2}-mu + lambda_{1} u + lambda_{2}v + mu uv}$
probability-theory statistics
edited Jan 29 at 14:25
YuiTo Cheng
2,1862937
$endgroup$
add a comment |
0
$begingroup$
Given the probability generating function of two random variables $(X,Y)$ as $G_{X,Y} (u,v)$, how to find the marginal densities of X and Y?
Specifying the probability generating function:
$G_{X,Y} (u,v) = e^{-lambda_{1} -lambda_{2}-mu + lambda_{1} u + lambda_{2}v + mu uv}$
probability-theory statistics
edited Jan 29 at 14:25
YuiTo Cheng
2,1862937
$endgroup$
$begingroup$
Note that $G_{X,Y}(u,v)=E(u^Xv^Y)$ hence the identities $G_{X,Y}(u,1)=E(u^X)=G_X(u)$ and $G_{X,Y}(1,v)=E(v^Y)=G_Y(v)$ give the marginals. For example, $$G_X(u)=e^{-(lambda_1+mu)(1-u)}$$ from which you should recognize the distribution of $X$.
$endgroup$
– Did
Jan 29 at 22:04
$begingroup$
Okay! Thank you.
$endgroup$
– user638402
Jan 30 at 7:26
$begingroup$
You are welcome. So, what is the distribution of $X$?
$endgroup$
– Did
Jan 30 at 7:35
add a comment |
0
0
0
$begingroup$
Given the probability generating function of two random variables $(X,Y)$ as $G_{X,Y} (u,v)$, how to find the marginal densities of X and Y?
Specifying the probability generating function:
$G_{X,Y} (u,v) = e^{-lambda_{1} -lambda_{2}-mu + lambda_{1} u + lambda_{2}v + mu uv}$
probability-theory statistics
edited Jan 29 at 14:25
YuiTo Cheng
2,1862937
$endgroup$
Given the probability generating function of two random variables $(X,Y)$ as $G_{X,Y} (u,v)$, how to find the marginal densities of X and Y?
Specifying the probability generating function:
$G_{X,Y} (u,v) = e^{-lambda_{1} -lambda_{2}-mu + lambda_{1} u + lambda_{2}v + mu uv}$
probability-theory statistics
probability-theory statistics
edited Jan 29 at 14:25
YuiTo Cheng
2,1862937
edited Jan 29 at 14:25
YuiTo Cheng
2,1862937
edited Jan 29 at 14:25
YuiTo Cheng
2,1862937
edited Jan 29 at 14:25
YuiTo Cheng
2,1862937
edited Jan 29 at 14:25
YuiTo Cheng
2,1862937
2,1862937
asked Jan 29 at 7:40
user638402
$begingroup$
Note that $G_{X,Y}(u,v)=E(u^Xv^Y)$ hence the identities $G_{X,Y}(u,1)=E(u^X)=G_X(u)$ and $G_{X,Y}(1,v)=E(v^Y)=G_Y(v)$ give the marginals. For example, $$G_X(u)=e^{-(lambda_1+mu)(1-u)}$$ from which you should recognize the distribution of $X$.
$endgroup$
– Did
Jan 29 at 22:04
$begingroup$
Okay! Thank you.
$endgroup$
– user638402
Jan 30 at 7:26
$begingroup$
You are welcome. So, what is the distribution of $X$?
$endgroup$
– Did
Jan 30 at 7:35
add a comment |
$begingroup$
Note that $G_{X,Y}(u,v)=E(u^Xv^Y)$ hence the identities $G_{X,Y}(u,1)=E(u^X)=G_X(u)$ and $G_{X,Y}(1,v)=E(v^Y)=G_Y(v)$ give the marginals. For example, $$G_X(u)=e^{-(lambda_1+mu)(1-u)}$$ from which you should recognize the distribution of $X$.
$endgroup$
– Did
Jan 29 at 22:04
$begingroup$
Okay! Thank you.
$endgroup$
– user638402
Jan 30 at 7:26
$begingroup$
You are welcome. So, what is the distribution of $X$?
$endgroup$
– Did
Jan 30 at 7:35
$begingroup$
Note that $G_{X,Y}(u,v)=E(u^Xv^Y)$ hence the identities $G_{X,Y}(u,1)=E(u^X)=G_X(u)$ and $G_{X,Y}(1,v)=E(v^Y)=G_Y(v)$ give the marginals. For example, $$G_X(u)=e^{-(lambda_1+mu)(1-u)}$$ from which you should recognize the distribution of $X$.
$endgroup$
– Did
Jan 29 at 22:04
$begingroup$
Note that $G_{X,Y}(u,v)=E(u^Xv^Y)$ hence the identities $G_{X,Y}(u,1)=E(u^X)=G_X(u)$ and $G_{X,Y}(1,v)=E(v^Y)=G_Y(v)$ give the marginals. For example, $$G_X(u)=e^{-(lambda_1+mu)(1-u)}$$ from which you should recognize the distribution of $X$.
$endgroup$
– Did
Jan 29 at 22:04
$begingroup$
Okay! Thank you.
$endgroup$
– user638402
Jan 30 at 7:26
$begingroup$
Okay! Thank you.
$endgroup$
– user638402
Jan 30 at 7:26
$begingroup$
You are welcome. So, what is the distribution of $X$?
$endgroup$
– Did
Jan 30 at 7:35
$begingroup$
You are welcome. So, what is the distribution of $X$?
$endgroup$
– Did
Jan 30 at 7:35
add a comment |
0
active
oldest
votes
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
});
}
});
draft saved
draft discarded
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%2f3091890%2ffinding-marginal-density-from-probability-generating-function-of-two-random-vari%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
draft saved
draft discarded
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.
draft saved
draft discarded
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%2f3091890%2ffinding-marginal-density-from-probability-generating-function-of-two-random-vari%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
$begingroup$
Note that $G_{X,Y}(u,v)=E(u^Xv^Y)$ hence the identities $G_{X,Y}(u,1)=E(u^X)=G_X(u)$ and $G_{X,Y}(1,v)=E(v^Y)=G_Y(v)$ give the marginals. For example, $$G_X(u)=e^{-(lambda_1+mu)(1-u)}$$ from which you should recognize the distribution of $X$.
$endgroup$
– Did
Jan 29 at 22:04
$begingroup$
Okay! Thank you.
$endgroup$
– user638402
Jan 30 at 7:26
$begingroup$
You are welcome. So, what is the distribution of $X$?
$endgroup$
– Did
Jan 30 at 7:35