Find the expected value and the variance of the time at which the plumber completes the project
This is a problem 8 from Chapter 8 "Introduction to probability" by Anderson:
Our faucet is broken, and a plumber has been called. The arrival time
of the plumber is uniformly distributed between 1pm and 7pm. The time
required to fix the broken faucet is then exponentially distributed
with mean 30 minutes. Supposing that the two times are independent,
find the expected value and the variance of the time at which the
plumber completes the project.
I've solved it, but I am not sure if I interpreted the statement of the problem (translated into the language of probability distribution) correctly. So let X be the arrival time of the plumber, then $Xsim Unif[1,7]$. Let Y be the time it takes to fix the broken faucet, then $Ysim Exp(2)$. Let Z be the time at which the plumber completes the project, then $Z=X+Y$. Now $E[Z]=E[X+Y]=E[X]+E[Y]=frac{1+7}{2}+frac{1}{2}=4.5$
$Var(Z) = text{(since X and Y are independent)} Var(X)+Var(Y) = frac{{(7-1)}^2}{12}+frac{1}{2^2}=3.25$
Are there any mistakes in my solution? Thanks.
probability proof-verification
asked Dec 5 '18 at 2:51
dxdydzdxdydz
1949
add a comment |
This is a problem 8 from Chapter 8 "Introduction to probability" by Anderson:
Our faucet is broken, and a plumber has been called. The arrival time
of the plumber is uniformly distributed between 1pm and 7pm. The time
required to fix the broken faucet is then exponentially distributed
with mean 30 minutes. Supposing that the two times are independent,
find the expected value and the variance of the time at which the
plumber completes the project.
I've solved it, but I am not sure if I interpreted the statement of the problem (translated into the language of probability distribution) correctly. So let X be the arrival time of the plumber, then $Xsim Unif[1,7]$. Let Y be the time it takes to fix the broken faucet, then $Ysim Exp(2)$. Let Z be the time at which the plumber completes the project, then $Z=X+Y$. Now $E[Z]=E[X+Y]=E[X]+E[Y]=frac{1+7}{2}+frac{1}{2}=4.5$
$Var(Z) = text{(since X and Y are independent)} Var(X)+Var(Y) = frac{{(7-1)}^2}{12}+frac{1}{2^2}=3.25$
Are there any mistakes in my solution? Thanks.
probability proof-verification
asked Dec 5 '18 at 2:51
dxdydzdxdydz
1949
add a comment |
This is a problem 8 from Chapter 8 "Introduction to probability" by Anderson:
Our faucet is broken, and a plumber has been called. The arrival time
of the plumber is uniformly distributed between 1pm and 7pm. The time
required to fix the broken faucet is then exponentially distributed
with mean 30 minutes. Supposing that the two times are independent,
find the expected value and the variance of the time at which the
plumber completes the project.
I've solved it, but I am not sure if I interpreted the statement of the problem (translated into the language of probability distribution) correctly. So let X be the arrival time of the plumber, then $Xsim Unif[1,7]$. Let Y be the time it takes to fix the broken faucet, then $Ysim Exp(2)$. Let Z be the time at which the plumber completes the project, then $Z=X+Y$. Now $E[Z]=E[X+Y]=E[X]+E[Y]=frac{1+7}{2}+frac{1}{2}=4.5$
$Var(Z) = text{(since X and Y are independent)} Var(X)+Var(Y) = frac{{(7-1)}^2}{12}+frac{1}{2^2}=3.25$
Are there any mistakes in my solution? Thanks.
probability proof-verification
asked Dec 5 '18 at 2:51
dxdydzdxdydz
1949
This is a problem 8 from Chapter 8 "Introduction to probability" by Anderson:
Our faucet is broken, and a plumber has been called. The arrival time
of the plumber is uniformly distributed between 1pm and 7pm. The time
required to fix the broken faucet is then exponentially distributed
with mean 30 minutes. Supposing that the two times are independent,
find the expected value and the variance of the time at which the
plumber completes the project.
I've solved it, but I am not sure if I interpreted the statement of the problem (translated into the language of probability distribution) correctly. So let X be the arrival time of the plumber, then $Xsim Unif[1,7]$. Let Y be the time it takes to fix the broken faucet, then $Ysim Exp(2)$. Let Z be the time at which the plumber completes the project, then $Z=X+Y$. Now $E[Z]=E[X+Y]=E[X]+E[Y]=frac{1+7}{2}+frac{1}{2}=4.5$
$Var(Z) = text{(since X and Y are independent)} Var(X)+Var(Y) = frac{{(7-1)}^2}{12}+frac{1}{2^2}=3.25$
Are there any mistakes in my solution? Thanks.
probability proof-verification
probability proof-verification
asked Dec 5 '18 at 2:51
dxdydzdxdydz
1949
asked Dec 5 '18 at 2:51
dxdydzdxdydz
1949
asked Dec 5 '18 at 2:51
dxdydzdxdydz
1949
asked Dec 5 '18 at 2:51
dxdydzdxdydz
1949
asked Dec 5 '18 at 2:51
dxdydzdxdydz
1949
1949
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Looks good. You have identified the distributions -and their mean and variance- correctly, and used the right equations -including the effect of independance-. If the calculations are okay, then you have it.
answered Dec 5 '18 at 2:59
Graham KempGraham Kemp
84.7k43378
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%2f3026531%2ffind-the-expected-value-and-the-variance-of-the-time-at-which-the-plumber-comple%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
Looks good. You have identified the distributions -and their mean and variance- correctly, and used the right equations -including the effect of independance-. If the calculations are okay, then you have it.
answered Dec 5 '18 at 2:59
Graham KempGraham Kemp
84.7k43378
add a comment |
Looks good. You have identified the distributions -and their mean and variance- correctly, and used the right equations -including the effect of independance-. If the calculations are okay, then you have it.
answered Dec 5 '18 at 2:59
Graham KempGraham Kemp
84.7k43378
add a comment |
Looks good. You have identified the distributions -and their mean and variance- correctly, and used the right equations -including the effect of independance-. If the calculations are okay, then you have it.
answered Dec 5 '18 at 2:59
Graham KempGraham Kemp
84.7k43378
Looks good. You have identified the distributions -and their mean and variance- correctly, and used the right equations -including the effect of independance-. If the calculations are okay, then you have it.
answered Dec 5 '18 at 2:59
Graham KempGraham Kemp
84.7k43378
answered Dec 5 '18 at 2:59
Graham KempGraham Kemp
84.7k43378
answered Dec 5 '18 at 2:59
Graham KempGraham Kemp
84.7k43378
answered Dec 5 '18 at 2:59
Graham KempGraham Kemp
84.7k43378
84.7k43378
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.
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%2f3026531%2ffind-the-expected-value-and-the-variance-of-the-time-at-which-the-plumber-comple%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
