Mixing
![Approximating probability of Cumulative Sum of random variables samples from Uniform[-1,1]](https://lh5.googleusercontent.com/-PBP73op69lo/AAAAAAAAAAI/AAAAAAAAUcQ/o9MZrYfCMak/s72-c/photo.jpg?sz=32)
Solution verification for a hypothesis testing question
$begingroup$
I am posting this question as a solution checking.
Let $X_1,...,X_{30}$ be a random sample from the exponential distribution with unknown mean $muin {1,1/delta}$ (where $delta>1)$. Consider the best test of $H_0:mu=1$ v.s. $H_1:mu=1/delta$ with power given by $beta=int_0^{17}dfrac {delta^{30}}{Gamma(30)}y^{29}e^{-delta y} dy$. What is the critical region of the test ?
My work: By Neyman-Pearson lemma, the best test is given by $L(1;mathbb x)/L(1/delta;mathbb x)le k$ i.e. $e^{-sum x_i}/e^{-delta sum x_i} le c$ i.e. $e^{(delta-1)sum x_i}le c $ i.e. $sum x_i le c_1$ for some constant $c_1$. Now given $P_{H_1}(sum_{i=1}^{30} X_i le c_1)=int_0^{17}dfrac {delta^{30}}{Gamma(30)}y^{29}e^{-delta y} dy$. Now under $H_1$, $X_i sim exp (delta)$ , so $sum_{i=1}^{30} X_i sim Gamma(30,1/delta)$, hence $c_1=17$.
Thus the critical region is $sum_{i=1}^{30} X_ile 17$.
Am I correct ?
Please let me know.
Thanks in advance
probability-theory statistics proof-verification probability-distributions hypothesis-testing
asked Jan 13 at 5:26
user521337user521337
1,1801416
$endgroup$
add a comment |
$begingroup$
I am posting this question as a solution checking.
Let $X_1,...,X_{30}$ be a random sample from the exponential distribution with unknown mean $muin {1,1/delta}$ (where $delta>1)$. Consider the best test of $H_0:mu=1$ v.s. $H_1:mu=1/delta$ with power given by $beta=int_0^{17}dfrac {delta^{30}}{Gamma(30)}y^{29}e^{-delta y} dy$. What is the critical region of the test ?
My work: By Neyman-Pearson lemma, the best test is given by $L(1;mathbb x)/L(1/delta;mathbb x)le k$ i.e. $e^{-sum x_i}/e^{-delta sum x_i} le c$ i.e. $e^{(delta-1)sum x_i}le c $ i.e. $sum x_i le c_1$ for some constant $c_1$. Now given $P_{H_1}(sum_{i=1}^{30} X_i le c_1)=int_0^{17}dfrac {delta^{30}}{Gamma(30)}y^{29}e^{-delta y} dy$. Now under $H_1$, $X_i sim exp (delta)$ , so $sum_{i=1}^{30} X_i sim Gamma(30,1/delta)$, hence $c_1=17$.
Thus the critical region is $sum_{i=1}^{30} X_ile 17$.
Am I correct ?
Please let me know.
Thanks in advance
probability-theory statistics proof-verification probability-distributions hypothesis-testing
asked Jan 13 at 5:26
user521337user521337
1,1801416
$endgroup$
1
$begingroup$
Please do not cross post across sites (both of your questions). It is against site rules. Decide where you want to ask them.
$endgroup$
– StubbornAtom
Jan 13 at 6:08
add a comment |
$begingroup$
I am posting this question as a solution checking.
Let $X_1,...,X_{30}$ be a random sample from the exponential distribution with unknown mean $muin {1,1/delta}$ (where $delta>1)$. Consider the best test of $H_0:mu=1$ v.s. $H_1:mu=1/delta$ with power given by $beta=int_0^{17}dfrac {delta^{30}}{Gamma(30)}y^{29}e^{-delta y} dy$. What is the critical region of the test ?
My work: By Neyman-Pearson lemma, the best test is given by $L(1;mathbb x)/L(1/delta;mathbb x)le k$ i.e. $e^{-sum x_i}/e^{-delta sum x_i} le c$ i.e. $e^{(delta-1)sum x_i}le c $ i.e. $sum x_i le c_1$ for some constant $c_1$. Now given $P_{H_1}(sum_{i=1}^{30} X_i le c_1)=int_0^{17}dfrac {delta^{30}}{Gamma(30)}y^{29}e^{-delta y} dy$. Now under $H_1$, $X_i sim exp (delta)$ , so $sum_{i=1}^{30} X_i sim Gamma(30,1/delta)$, hence $c_1=17$.
Thus the critical region is $sum_{i=1}^{30} X_ile 17$.
Am I correct ?
Please let me know.
Thanks in advance
probability-theory statistics proof-verification probability-distributions hypothesis-testing
asked Jan 13 at 5:26
user521337user521337
1,1801416
$endgroup$
I am posting this question as a solution checking.
Let $X_1,...,X_{30}$ be a random sample from the exponential distribution with unknown mean $muin {1,1/delta}$ (where $delta>1)$. Consider the best test of $H_0:mu=1$ v.s. $H_1:mu=1/delta$ with power given by $beta=int_0^{17}dfrac {delta^{30}}{Gamma(30)}y^{29}e^{-delta y} dy$. What is the critical region of the test ?
My work: By Neyman-Pearson lemma, the best test is given by $L(1;mathbb x)/L(1/delta;mathbb x)le k$ i.e. $e^{-sum x_i}/e^{-delta sum x_i} le c$ i.e. $e^{(delta-1)sum x_i}le c $ i.e. $sum x_i le c_1$ for some constant $c_1$. Now given $P_{H_1}(sum_{i=1}^{30} X_i le c_1)=int_0^{17}dfrac {delta^{30}}{Gamma(30)}y^{29}e^{-delta y} dy$. Now under $H_1$, $X_i sim exp (delta)$ , so $sum_{i=1}^{30} X_i sim Gamma(30,1/delta)$, hence $c_1=17$.
Thus the critical region is $sum_{i=1}^{30} X_ile 17$.
Am I correct ?
Please let me know.
Thanks in advance
probability-theory statistics proof-verification probability-distributions hypothesis-testing
probability-theory statistics proof-verification probability-distributions hypothesis-testing
asked Jan 13 at 5:26
user521337user521337
1,1801416
asked Jan 13 at 5:26
user521337user521337
1,1801416
asked Jan 13 at 5:26
user521337user521337
1,1801416
asked Jan 13 at 5:26
user521337user521337
1,1801416
asked Jan 13 at 5:26
user521337user521337
1,1801416
1,1801416
1
$begingroup$
Please do not cross post across sites (both of your questions). It is against site rules. Decide where you want to ask them.
$endgroup$
– StubbornAtom
Jan 13 at 6:08
add a comment |
1
$begingroup$
Please do not cross post across sites (both of your questions). It is against site rules. Decide where you want to ask them.
$endgroup$
– StubbornAtom
Jan 13 at 6:08
1
1
$begingroup$
Please do not cross post across sites (both of your questions). It is against site rules. Decide where you want to ask them.
$endgroup$
– StubbornAtom
Jan 13 at 6:08
$begingroup$
Please do not cross post across sites (both of your questions). It is against site rules. Decide where you want to ask them.
$endgroup$
– StubbornAtom
Jan 13 at 6:08
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%2f3071727%2fsolution-verification-for-a-hypothesis-testing-question%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.
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%2f3071727%2fsolution-verification-for-a-hypothesis-testing-question%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
$begingroup$
Please do not cross post across sites (both of your questions). It is against site rules. Decide where you want to ask them.
$endgroup$
– StubbornAtom
Jan 13 at 6:08