Definition of Probability mass function of a random process.
A random process $X(t)$ is defined as $X(t)=1, 0leq t leq Y$ and $0$ otherwise where $Y$ follows an exponential distribution. What is the $pmf$ of $X(t)$
I am a bit confused on the definition of $pmf$ for random process. By definition, the pmf of $X(t)$ is
$$Pr({ X(t, xi)=1 })=Pr({ (t, xi) | 0leq t leq xi })=int_0^{infty}int_0^xi f(t,xi)dtdxi$$
However, I am confused how I can calculate $f(t, xi)$. I know from the question that $f(xi) = lambda e^{-lambda xi}$ where $lambda$ is the parameter of the geometric distribution. But I don't see how to get $f(t,xi)$
stochastic-processes
asked Nov 27 at 8:58
AspiringMat
545518
add a comment |
A random process $X(t)$ is defined as $X(t)=1, 0leq t leq Y$ and $0$ otherwise where $Y$ follows an exponential distribution. What is the $pmf$ of $X(t)$
I am a bit confused on the definition of $pmf$ for random process. By definition, the pmf of $X(t)$ is
$$Pr({ X(t, xi)=1 })=Pr({ (t, xi) | 0leq t leq xi })=int_0^{infty}int_0^xi f(t,xi)dtdxi$$
However, I am confused how I can calculate $f(t, xi)$. I know from the question that $f(xi) = lambda e^{-lambda xi}$ where $lambda$ is the parameter of the geometric distribution. But I don't see how to get $f(t,xi)$
stochastic-processes
asked Nov 27 at 8:58
AspiringMat
545518
My guess is that the question asks you to find the distribution of $X(t)$ for a fixed $t$ (and this is trivial). The distribution of the entire process is a measure on an infinite dimensional function space.
– Kavi Rama Murthy
Nov 27 at 9:13
@KaviRamaMurthy That makes a lot more sense now. Just integrate from $t$ to infinity the geometric pdf. This is one of the reason I don't like the notation of suppressing the sample space in random processes. It just makes things more confusing for learners such as myself. Thank you!
– AspiringMat
Nov 27 at 9:22
add a comment |
A random process $X(t)$ is defined as $X(t)=1, 0leq t leq Y$ and $0$ otherwise where $Y$ follows an exponential distribution. What is the $pmf$ of $X(t)$
I am a bit confused on the definition of $pmf$ for random process. By definition, the pmf of $X(t)$ is
$$Pr({ X(t, xi)=1 })=Pr({ (t, xi) | 0leq t leq xi })=int_0^{infty}int_0^xi f(t,xi)dtdxi$$
However, I am confused how I can calculate $f(t, xi)$. I know from the question that $f(xi) = lambda e^{-lambda xi}$ where $lambda$ is the parameter of the geometric distribution. But I don't see how to get $f(t,xi)$
stochastic-processes
asked Nov 27 at 8:58
AspiringMat
545518
A random process $X(t)$ is defined as $X(t)=1, 0leq t leq Y$ and $0$ otherwise where $Y$ follows an exponential distribution. What is the $pmf$ of $X(t)$
I am a bit confused on the definition of $pmf$ for random process. By definition, the pmf of $X(t)$ is
$$Pr({ X(t, xi)=1 })=Pr({ (t, xi) | 0leq t leq xi })=int_0^{infty}int_0^xi f(t,xi)dtdxi$$
However, I am confused how I can calculate $f(t, xi)$. I know from the question that $f(xi) = lambda e^{-lambda xi}$ where $lambda$ is the parameter of the geometric distribution. But I don't see how to get $f(t,xi)$
stochastic-processes
stochastic-processes
asked Nov 27 at 8:58
AspiringMat
545518
asked Nov 27 at 8:58
AspiringMat
545518
edited Nov 27 at 9:09
asked Nov 27 at 8:58
AspiringMat
545518
asked Nov 27 at 8:58
AspiringMat
545518
asked Nov 27 at 8:58
AspiringMat
545518
545518
My guess is that the question asks you to find the distribution of $X(t)$ for a fixed $t$ (and this is trivial). The distribution of the entire process is a measure on an infinite dimensional function space.
– Kavi Rama Murthy
Nov 27 at 9:13
@KaviRamaMurthy That makes a lot more sense now. Just integrate from $t$ to infinity the geometric pdf. This is one of the reason I don't like the notation of suppressing the sample space in random processes. It just makes things more confusing for learners such as myself. Thank you!
– AspiringMat
Nov 27 at 9:22
add a comment |
My guess is that the question asks you to find the distribution of $X(t)$ for a fixed $t$ (and this is trivial). The distribution of the entire process is a measure on an infinite dimensional function space.
– Kavi Rama Murthy
Nov 27 at 9:13
@KaviRamaMurthy That makes a lot more sense now. Just integrate from $t$ to infinity the geometric pdf. This is one of the reason I don't like the notation of suppressing the sample space in random processes. It just makes things more confusing for learners such as myself. Thank you!
– AspiringMat
Nov 27 at 9:22
My guess is that the question asks you to find the distribution of $X(t)$ for a fixed $t$ (and this is trivial). The distribution of the entire process is a measure on an infinite dimensional function space.
– Kavi Rama Murthy
Nov 27 at 9:13
My guess is that the question asks you to find the distribution of $X(t)$ for a fixed $t$ (and this is trivial). The distribution of the entire process is a measure on an infinite dimensional function space.
– Kavi Rama Murthy
Nov 27 at 9:13
@KaviRamaMurthy That makes a lot more sense now. Just integrate from $t$ to infinity the geometric pdf. This is one of the reason I don't like the notation of suppressing the sample space in random processes. It just makes things more confusing for learners such as myself. Thank you!
– AspiringMat
Nov 27 at 9:22
@KaviRamaMurthy That makes a lot more sense now. Just integrate from $t$ to infinity the geometric pdf. This is one of the reason I don't like the notation of suppressing the sample space in random processes. It just makes things more confusing for learners such as myself. Thank you!
– AspiringMat
Nov 27 at 9:22
add a comment |
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%2f3015526%2fdefinition-of-probability-mass-function-of-a-random-process%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
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%2f3015526%2fdefinition-of-probability-mass-function-of-a-random-process%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
My guess is that the question asks you to find the distribution of $X(t)$ for a fixed $t$ (and this is trivial). The distribution of the entire process is a measure on an infinite dimensional function space.
– Kavi Rama Murthy
Nov 27 at 9:13
@KaviRamaMurthy That makes a lot more sense now. Just integrate from $t$ to infinity the geometric pdf. This is one of the reason I don't like the notation of suppressing the sample space in random processes. It just makes things more confusing for learners such as myself. Thank you!
– AspiringMat
Nov 27 at 9:22