Markov Chain for a teleportation machine
$begingroup$
I am given the following problem:
Every time I teleport, the teleportation machine decides to transport me to the very same place I am with probability $α ∈ (0,1)$, or, if I don't repeat the same location, it randomly select one of the other $N−1$ places to which I can teleport and places me there instead. Find the limiting probability
$$
lim_{n→∞}
P(text{nth teleportation place is the same as the first place})
$$
I am also given the following hint: Define a suitable two state Markov Chain with states {initial place, other place}.
I approach the problem in the following way: Let $X_n = 0$ be the state in which we are at the initial place. Let $X_n = 1$ be the state in which we are at other place. Then the transition matrix is partially given by
$$
P=
begin{bmatrix}
alpha & 1-alpha \
beta & 1-beta
end{bmatrix}
$$
However, I can't seem to find $beta$. I tried used Bayes' rule and conditional probability but to no avail. Any help would be greatly appreciated
probability markov-chains
asked Jan 15 at 8:45
Mr_RobotMr_Robot
486
$endgroup$
add a comment |
$begingroup$
I am given the following problem:
Every time I teleport, the teleportation machine decides to transport me to the very same place I am with probability $α ∈ (0,1)$, or, if I don't repeat the same location, it randomly select one of the other $N−1$ places to which I can teleport and places me there instead. Find the limiting probability
$$
lim_{n→∞}
P(text{nth teleportation place is the same as the first place})
$$
I am also given the following hint: Define a suitable two state Markov Chain with states {initial place, other place}.
I approach the problem in the following way: Let $X_n = 0$ be the state in which we are at the initial place. Let $X_n = 1$ be the state in which we are at other place. Then the transition matrix is partially given by
$$
P=
begin{bmatrix}
alpha & 1-alpha \
beta & 1-beta
end{bmatrix}
$$
However, I can't seem to find $beta$. I tried used Bayes' rule and conditional probability but to no avail. Any help would be greatly appreciated
probability markov-chains
asked Jan 15 at 8:45
Mr_RobotMr_Robot
486
$endgroup$
add a comment |
$begingroup$
I am given the following problem:
Every time I teleport, the teleportation machine decides to transport me to the very same place I am with probability $α ∈ (0,1)$, or, if I don't repeat the same location, it randomly select one of the other $N−1$ places to which I can teleport and places me there instead. Find the limiting probability
$$
lim_{n→∞}
P(text{nth teleportation place is the same as the first place})
$$
I am also given the following hint: Define a suitable two state Markov Chain with states {initial place, other place}.
I approach the problem in the following way: Let $X_n = 0$ be the state in which we are at the initial place. Let $X_n = 1$ be the state in which we are at other place. Then the transition matrix is partially given by
$$
P=
begin{bmatrix}
alpha & 1-alpha \
beta & 1-beta
end{bmatrix}
$$
However, I can't seem to find $beta$. I tried used Bayes' rule and conditional probability but to no avail. Any help would be greatly appreciated
probability markov-chains
asked Jan 15 at 8:45
Mr_RobotMr_Robot
486
$endgroup$
I am given the following problem:
Every time I teleport, the teleportation machine decides to transport me to the very same place I am with probability $α ∈ (0,1)$, or, if I don't repeat the same location, it randomly select one of the other $N−1$ places to which I can teleport and places me there instead. Find the limiting probability
$$
lim_{n→∞}
P(text{nth teleportation place is the same as the first place})
$$
I am also given the following hint: Define a suitable two state Markov Chain with states {initial place, other place}.
I approach the problem in the following way: Let $X_n = 0$ be the state in which we are at the initial place. Let $X_n = 1$ be the state in which we are at other place. Then the transition matrix is partially given by
$$
P=
begin{bmatrix}
alpha & 1-alpha \
beta & 1-beta
end{bmatrix}
$$
However, I can't seem to find $beta$. I tried used Bayes' rule and conditional probability but to no avail. Any help would be greatly appreciated
probability markov-chains
probability markov-chains
asked Jan 15 at 8:45
Mr_RobotMr_Robot
486
asked Jan 15 at 8:45
Mr_RobotMr_Robot
486
asked Jan 15 at 8:45
Mr_RobotMr_Robot
486
asked Jan 15 at 8:45
Mr_RobotMr_Robot
486
asked Jan 15 at 8:45
Mr_RobotMr_Robot
486
486
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If you are in another place, the probability you come back to the initial place is
the probability you do not stay still $(1-alpha)$ multiplied by the probability the initial place is chosen $left(frac{1}{n-1}right)$, so
$$beta = frac{1-alpha}{n-1}$$
answered Jan 15 at 8:49
HenryHenry
99.9k480165
$endgroup$
add a comment |
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%2f3074206%2fmarkov-chain-for-a-teleportation-machine%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If you are in another place, the probability you come back to the initial place is
the probability you do not stay still $(1-alpha)$ multiplied by the probability the initial place is chosen $left(frac{1}{n-1}right)$, so
$$beta = frac{1-alpha}{n-1}$$
answered Jan 15 at 8:49
HenryHenry
99.9k480165
$endgroup$
add a comment |
$begingroup$
If you are in another place, the probability you come back to the initial place is
the probability you do not stay still $(1-alpha)$ multiplied by the probability the initial place is chosen $left(frac{1}{n-1}right)$, so
$$beta = frac{1-alpha}{n-1}$$
answered Jan 15 at 8:49
HenryHenry
99.9k480165
$endgroup$
add a comment |
$begingroup$
If you are in another place, the probability you come back to the initial place is
the probability you do not stay still $(1-alpha)$ multiplied by the probability the initial place is chosen $left(frac{1}{n-1}right)$, so
$$beta = frac{1-alpha}{n-1}$$
answered Jan 15 at 8:49
HenryHenry
99.9k480165
$endgroup$
If you are in another place, the probability you come back to the initial place is
the probability you do not stay still $(1-alpha)$ multiplied by the probability the initial place is chosen $left(frac{1}{n-1}right)$, so
$$beta = frac{1-alpha}{n-1}$$
answered Jan 15 at 8:49
HenryHenry
99.9k480165
answered Jan 15 at 8:49
HenryHenry
99.9k480165
answered Jan 15 at 8:49
HenryHenry
99.9k480165
answered Jan 15 at 8:49
HenryHenry
99.9k480165
99.9k480165
add a comment |
add a comment |
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%2f3074206%2fmarkov-chain-for-a-teleportation-machine%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
