Probability of a machine working at certain time
This question arises while I am learning Continuous Time Markov Chain :
A machine is working for an exponential time with rate $mu$ before breaking
down. The repair time of the machine is exponentially distributed with rate $lambda$. Find the probability that the machine is working at a certain time $T$.
Note : The pdf of an exponentially distributed random variable with rate $lambda$ is $lambda e^{-lambda x}$
My idea is to conditioning on what time the machine starts breaking down and all I get is
$$ P[text{Machine working at time $T$}] = int_0^T P[text{Machine working at time T} | text{Machine is breaking down at $x$}] mu e^{-mu x} ~ dx $$
and I couldn't proceed any further.
Any ideas on how to solve it? Thanks!
statistics markov-chains exponential-distribution
asked Dec 9 at 8:38
Ricky The Ising
414
add a comment |
This question arises while I am learning Continuous Time Markov Chain :
A machine is working for an exponential time with rate $mu$ before breaking
down. The repair time of the machine is exponentially distributed with rate $lambda$. Find the probability that the machine is working at a certain time $T$.
Note : The pdf of an exponentially distributed random variable with rate $lambda$ is $lambda e^{-lambda x}$
My idea is to conditioning on what time the machine starts breaking down and all I get is
$$ P[text{Machine working at time $T$}] = int_0^T P[text{Machine working at time T} | text{Machine is breaking down at $x$}] mu e^{-mu x} ~ dx $$
and I couldn't proceed any further.
Any ideas on how to solve it? Thanks!
statistics markov-chains exponential-distribution
asked Dec 9 at 8:38
Ricky The Ising
414
add a comment |
This question arises while I am learning Continuous Time Markov Chain :
A machine is working for an exponential time with rate $mu$ before breaking
down. The repair time of the machine is exponentially distributed with rate $lambda$. Find the probability that the machine is working at a certain time $T$.
Note : The pdf of an exponentially distributed random variable with rate $lambda$ is $lambda e^{-lambda x}$
My idea is to conditioning on what time the machine starts breaking down and all I get is
$$ P[text{Machine working at time $T$}] = int_0^T P[text{Machine working at time T} | text{Machine is breaking down at $x$}] mu e^{-mu x} ~ dx $$
and I couldn't proceed any further.
Any ideas on how to solve it? Thanks!
statistics markov-chains exponential-distribution
asked Dec 9 at 8:38
Ricky The Ising
414
This question arises while I am learning Continuous Time Markov Chain :
A machine is working for an exponential time with rate $mu$ before breaking
down. The repair time of the machine is exponentially distributed with rate $lambda$. Find the probability that the machine is working at a certain time $T$.
Note : The pdf of an exponentially distributed random variable with rate $lambda$ is $lambda e^{-lambda x}$
My idea is to conditioning on what time the machine starts breaking down and all I get is
$$ P[text{Machine working at time $T$}] = int_0^T P[text{Machine working at time T} | text{Machine is breaking down at $x$}] mu e^{-mu x} ~ dx $$
and I couldn't proceed any further.
Any ideas on how to solve it? Thanks!
statistics markov-chains exponential-distribution
statistics markov-chains exponential-distribution
asked Dec 9 at 8:38
Ricky The Ising
414
asked Dec 9 at 8:38
Ricky The Ising
414
asked Dec 9 at 8:38
Ricky The Ising
414
asked Dec 9 at 8:38
Ricky The Ising
414
asked Dec 9 at 8:38
Ricky The Ising
414
414
add a comment |
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%2f3032162%2fprobability-of-a-machine-working-at-certain-time%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%2f3032162%2fprobability-of-a-machine-working-at-certain-time%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
