Proof of a two-dimensional random variable
A two-dimensional random variable $ (X,Y) $ has total probability density function like $ f _{X,Y} (x,y) $.
And I want to prove that probability density function of a random variable $Z=X+Y$
is $f_{Z}(z) = int_{- infty }^{ infty } f _{X,Y} (x, z-x)dx$ .
And I don't know how to do it? Any help.
probability
edited Dec 28 '18 at 9:46
Hayk
2,0721213
asked Dec 28 '18 at 8:51
Abakus
32
add a comment |
A two-dimensional random variable $ (X,Y) $ has total probability density function like $ f _{X,Y} (x,y) $.
And I want to prove that probability density function of a random variable $Z=X+Y$
is $f_{Z}(z) = int_{- infty }^{ infty } f _{X,Y} (x, z-x)dx$ .
And I don't know how to do it? Any help.
probability
edited Dec 28 '18 at 9:46
Hayk
2,0721213
asked Dec 28 '18 at 8:51
Abakus
32
1
Start with $P[Z leq z] =...$ and differentiate with respect to $z$. You can either do a 2-d integral to compute $P[Zleq z]$, or teh law of total probability.
– Michael
Dec 28 '18 at 8:57
Apply Fubini's Theorem.
– Kavi Rama Murthy
Dec 28 '18 at 9:26
add a comment |
A two-dimensional random variable $ (X,Y) $ has total probability density function like $ f _{X,Y} (x,y) $.
And I want to prove that probability density function of a random variable $Z=X+Y$
is $f_{Z}(z) = int_{- infty }^{ infty } f _{X,Y} (x, z-x)dx$ .
And I don't know how to do it? Any help.
probability
edited Dec 28 '18 at 9:46
Hayk
2,0721213
asked Dec 28 '18 at 8:51
Abakus
32
A two-dimensional random variable $ (X,Y) $ has total probability density function like $ f _{X,Y} (x,y) $.
And I want to prove that probability density function of a random variable $Z=X+Y$
is $f_{Z}(z) = int_{- infty }^{ infty } f _{X,Y} (x, z-x)dx$ .
And I don't know how to do it? Any help.
probability
probability
edited Dec 28 '18 at 9:46
Hayk
2,0721213
asked Dec 28 '18 at 8:51
Abakus
32
edited Dec 28 '18 at 9:46
Hayk
2,0721213
asked Dec 28 '18 at 8:51
Abakus
32
edited Dec 28 '18 at 9:46
Hayk
2,0721213
edited Dec 28 '18 at 9:46
Hayk
2,0721213
edited Dec 28 '18 at 9:46
Hayk
2,0721213
2,0721213
asked Dec 28 '18 at 8:51
Abakus
32
asked Dec 28 '18 at 8:51
Abakus
32
asked Dec 28 '18 at 8:51
Abakus
32
32
1
Start with $P[Z leq z] =...$ and differentiate with respect to $z$. You can either do a 2-d integral to compute $P[Zleq z]$, or teh law of total probability.
– Michael
Dec 28 '18 at 8:57
Apply Fubini's Theorem.
– Kavi Rama Murthy
Dec 28 '18 at 9:26
add a comment |
1
Start with $P[Z leq z] =...$ and differentiate with respect to $z$. You can either do a 2-d integral to compute $P[Zleq z]$, or teh law of total probability.
– Michael
Dec 28 '18 at 8:57
Apply Fubini's Theorem.
– Kavi Rama Murthy
Dec 28 '18 at 9:26
1
1
Start with $P[Z leq z] =...$ and differentiate with respect to $z$. You can either do a 2-d integral to compute $P[Zleq z]$, or teh law of total probability.
– Michael
Dec 28 '18 at 8:57
Start with $P[Z leq z] =...$ and differentiate with respect to $z$. You can either do a 2-d integral to compute $P[Zleq z]$, or teh law of total probability.
– Michael
Dec 28 '18 at 8:57
Apply Fubini's Theorem.
– Kavi Rama Murthy
Dec 28 '18 at 9:26
Apply Fubini's Theorem.
– Kavi Rama Murthy
Dec 28 '18 at 9:26
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%2f3054691%2fproof-of-a-two-dimensional-random-variable%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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3054691%2fproof-of-a-two-dimensional-random-variable%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
1
Start with $P[Z leq z] =...$ and differentiate with respect to $z$. You can either do a 2-d integral to compute $P[Zleq z]$, or teh law of total probability.
– Michael
Dec 28 '18 at 8:57
Apply Fubini's Theorem.
– Kavi Rama Murthy
Dec 28 '18 at 9:26