Finding a probability density from an exponential family using a moment generating function Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Finding a probability distribution given the moment generating functionHow to find probability distribution function given the Moment Generating FunctionProbability density function with the help of the Laplace (Fourier) transformCompute conditional expectation from moment generating functionFind the moment generating function of the random variableFinding Probability Density Function and ProbabilityExpectation and Variance using Moment Generating FunctionsMoment generating function within another functionFinding the probability density function of $Z = X + Y$Establish bound for a probability using moment generating function
Is it possible for SQL statements to execute concurrently within a single session in SQL Server?
Crossing US/Canada Border for less than 24 hours
What is the font for "b" letter?
What is the difference between globalisation and imperialism?
Is there a kind of relay only consumes power when switching?
Is it fair for a professor to grade us on the possession of past papers?
Should I follow up with an employee I believe overracted to a mistake I made?
Project Euler #1 in C++
How to tell that you are a giant?
What does it mean that physics no longer uses mechanical models to describe phenomena?
How does light 'choose' between wave and particle behaviour?
NumericArray versus PackedArray in MMA12
Is there any word for a place full of confusion?
Drawing without replacement: why is the order of draw irrelevant?
What is the topology associated with the algebras for the ultrafilter monad?
Did Deadpool rescue all of the X-Force?
What do you call the main part of a joke?
Why do we need to use the builder design pattern when we can do the same thing with setters?
Should I use a zero-interest credit card for a large one-time purchase?
Most bit efficient text communication method?
How do I find out the mythology and history of my Fortress?
Putting class ranking in CV, but against dept guidelines
What initially awakened the Balrog?
Is grep documentation about ignoring case wrong, since it doesn't ignore case in filenames?
Finding a probability density from an exponential family using a moment generating function
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Finding a probability distribution given the moment generating functionHow to find probability distribution function given the Moment Generating FunctionProbability density function with the help of the Laplace (Fourier) transformCompute conditional expectation from moment generating functionFind the moment generating function of the random variableFinding Probability Density Function and ProbabilityExpectation and Variance using Moment Generating FunctionsMoment generating function within another functionFinding the probability density function of $Z = X + Y$Establish bound for a probability using moment generating function
$begingroup$
I would like to find the density of a sum of i.i.d. random variables $barY=frac1nu(Y_1+...+Y_nu)$, where the density of these random variables is $f_Y(y;theta)=e^theta y - b(theta)f_0(y)$. In turn, the density $f_0(y)$ is specified using the m.g.f. $M_0(xi)=e^b(xi)$ and a differentiable function $b(xi)$.
Edit: Found an additional bit of information that the resulting distribution has to be from the exponential family $f_barY(y;theta, phi)= exp(fracytheta-b(theta)a(phi)+c(y, phi))$, where $a(phi)=nu^-1$.
Attempt 1:
Omitting the details, I found the m.g.f. of $f_Y(y,theta)$ to be equal to $M_Y(xi)=e^b(xi + theta)-b(theta)$.
Then, I computed the m.g.f. of the sum of $Y_i$'s using this property, which resulted in
$$M_barY(xi)=left(M_Y(frac1nuxi)right)^nu=e^nuleft(b(frac1nuxi+theta)-b(theta)right)$$
I cannot find a way to transform this into $f_hatY(y; theta)$ and would greatly appreciate a pointer or a suggestion on how to proceed.
Attempt 2:
What seems like a possible solution is to recast $M_barY$ into the form of $M_Y$ and reconstruct the density from that. For example, if one were to introduce substitutions like $tildeb(theta)=nu b(theta)$ and $tildexi=frac1nuxi$, then the expressions of the two m.g.f.'s would be identical. This means that
$$M_barY(tildexi)=
int_-infty^inftye^tildexiye^theta y-tildeb(theta)f_0(y)dy=
int_-infty^inftye^tildexiye^theta y-nu b(theta)f_0(y)dy.$$
I am not entirely sure how to deal with $tildexi$ at this point... Is it a permissible operation to transform $y$ in the integral into $tildey=frac1nu y$? It would then follow that
$$int_-infty^inftye^tildexiye^theta y-nu b(theta)f_0(y)dy=
int_-infty^inftye^xitildeye^nutheta tildey-nu b(theta)nuf_0(nutildey)dtildey=
int_-infty^inftye^xitildeye^nutheta tildey-nu b(theta)+ln(nuf_0(nutildey))dtildey.$$
The density would then be expressed as $f_hatY(y;theta, nu)=e^nu(theta tildey-b(theta))+ln(nuf_0(nutildey))$ and it would have the form in the desired form mentioned in the Edit above.
Would this make sense? My math background is not really so strong, so I have a feeling that I could have made some illegitimate operations here...
probability probability-theory moment-generating-functions
asked Apr 1 at 18:00
J.K.J.K.
1637
$endgroup$
add a comment |
$begingroup$
I would like to find the density of a sum of i.i.d. random variables $barY=frac1nu(Y_1+...+Y_nu)$, where the density of these random variables is $f_Y(y;theta)=e^theta y - b(theta)f_0(y)$. In turn, the density $f_0(y)$ is specified using the m.g.f. $M_0(xi)=e^b(xi)$ and a differentiable function $b(xi)$.
Edit: Found an additional bit of information that the resulting distribution has to be from the exponential family $f_barY(y;theta, phi)= exp(fracytheta-b(theta)a(phi)+c(y, phi))$, where $a(phi)=nu^-1$.
Attempt 1:
Omitting the details, I found the m.g.f. of $f_Y(y,theta)$ to be equal to $M_Y(xi)=e^b(xi + theta)-b(theta)$.
Then, I computed the m.g.f. of the sum of $Y_i$'s using this property, which resulted in
$$M_barY(xi)=left(M_Y(frac1nuxi)right)^nu=e^nuleft(b(frac1nuxi+theta)-b(theta)right)$$
I cannot find a way to transform this into $f_hatY(y; theta)$ and would greatly appreciate a pointer or a suggestion on how to proceed.
Attempt 2:
What seems like a possible solution is to recast $M_barY$ into the form of $M_Y$ and reconstruct the density from that. For example, if one were to introduce substitutions like $tildeb(theta)=nu b(theta)$ and $tildexi=frac1nuxi$, then the expressions of the two m.g.f.'s would be identical. This means that
$$M_barY(tildexi)=
int_-infty^inftye^tildexiye^theta y-tildeb(theta)f_0(y)dy=
int_-infty^inftye^tildexiye^theta y-nu b(theta)f_0(y)dy.$$
I am not entirely sure how to deal with $tildexi$ at this point... Is it a permissible operation to transform $y$ in the integral into $tildey=frac1nu y$? It would then follow that
$$int_-infty^inftye^tildexiye^theta y-nu b(theta)f_0(y)dy=
int_-infty^inftye^xitildeye^nutheta tildey-nu b(theta)nuf_0(nutildey)dtildey=
int_-infty^inftye^xitildeye^nutheta tildey-nu b(theta)+ln(nuf_0(nutildey))dtildey.$$
The density would then be expressed as $f_hatY(y;theta, nu)=e^nu(theta tildey-b(theta))+ln(nuf_0(nutildey))$ and it would have the form in the desired form mentioned in the Edit above.
Would this make sense? My math background is not really so strong, so I have a feeling that I could have made some illegitimate operations here...
probability probability-theory moment-generating-functions
asked Apr 1 at 18:00
J.K.J.K.
1637
$endgroup$
add a comment |
$begingroup$
I would like to find the density of a sum of i.i.d. random variables $barY=frac1nu(Y_1+...+Y_nu)$, where the density of these random variables is $f_Y(y;theta)=e^theta y - b(theta)f_0(y)$. In turn, the density $f_0(y)$ is specified using the m.g.f. $M_0(xi)=e^b(xi)$ and a differentiable function $b(xi)$.
Edit: Found an additional bit of information that the resulting distribution has to be from the exponential family $f_barY(y;theta, phi)= exp(fracytheta-b(theta)a(phi)+c(y, phi))$, where $a(phi)=nu^-1$.
Attempt 1:
Omitting the details, I found the m.g.f. of $f_Y(y,theta)$ to be equal to $M_Y(xi)=e^b(xi + theta)-b(theta)$.
Then, I computed the m.g.f. of the sum of $Y_i$'s using this property, which resulted in
$$M_barY(xi)=left(M_Y(frac1nuxi)right)^nu=e^nuleft(b(frac1nuxi+theta)-b(theta)right)$$
I cannot find a way to transform this into $f_hatY(y; theta)$ and would greatly appreciate a pointer or a suggestion on how to proceed.
Attempt 2:
What seems like a possible solution is to recast $M_barY$ into the form of $M_Y$ and reconstruct the density from that. For example, if one were to introduce substitutions like $tildeb(theta)=nu b(theta)$ and $tildexi=frac1nuxi$, then the expressions of the two m.g.f.'s would be identical. This means that
$$M_barY(tildexi)=
int_-infty^inftye^tildexiye^theta y-tildeb(theta)f_0(y)dy=
int_-infty^inftye^tildexiye^theta y-nu b(theta)f_0(y)dy.$$
I am not entirely sure how to deal with $tildexi$ at this point... Is it a permissible operation to transform $y$ in the integral into $tildey=frac1nu y$? It would then follow that
$$int_-infty^inftye^tildexiye^theta y-nu b(theta)f_0(y)dy=
int_-infty^inftye^xitildeye^nutheta tildey-nu b(theta)nuf_0(nutildey)dtildey=
int_-infty^inftye^xitildeye^nutheta tildey-nu b(theta)+ln(nuf_0(nutildey))dtildey.$$
The density would then be expressed as $f_hatY(y;theta, nu)=e^nu(theta tildey-b(theta))+ln(nuf_0(nutildey))$ and it would have the form in the desired form mentioned in the Edit above.
Would this make sense? My math background is not really so strong, so I have a feeling that I could have made some illegitimate operations here...
probability probability-theory moment-generating-functions
asked Apr 1 at 18:00
J.K.J.K.
1637
$endgroup$
I would like to find the density of a sum of i.i.d. random variables $barY=frac1nu(Y_1+...+Y_nu)$, where the density of these random variables is $f_Y(y;theta)=e^theta y - b(theta)f_0(y)$. In turn, the density $f_0(y)$ is specified using the m.g.f. $M_0(xi)=e^b(xi)$ and a differentiable function $b(xi)$.
Edit: Found an additional bit of information that the resulting distribution has to be from the exponential family $f_barY(y;theta, phi)= exp(fracytheta-b(theta)a(phi)+c(y, phi))$, where $a(phi)=nu^-1$.
Attempt 1:
Omitting the details, I found the m.g.f. of $f_Y(y,theta)$ to be equal to $M_Y(xi)=e^b(xi + theta)-b(theta)$.
Then, I computed the m.g.f. of the sum of $Y_i$'s using this property, which resulted in
$$M_barY(xi)=left(M_Y(frac1nuxi)right)^nu=e^nuleft(b(frac1nuxi+theta)-b(theta)right)$$
I cannot find a way to transform this into $f_hatY(y; theta)$ and would greatly appreciate a pointer or a suggestion on how to proceed.
Attempt 2:
What seems like a possible solution is to recast $M_barY$ into the form of $M_Y$ and reconstruct the density from that. For example, if one were to introduce substitutions like $tildeb(theta)=nu b(theta)$ and $tildexi=frac1nuxi$, then the expressions of the two m.g.f.'s would be identical. This means that
$$M_barY(tildexi)=
int_-infty^inftye^tildexiye^theta y-tildeb(theta)f_0(y)dy=
int_-infty^inftye^tildexiye^theta y-nu b(theta)f_0(y)dy.$$
I am not entirely sure how to deal with $tildexi$ at this point... Is it a permissible operation to transform $y$ in the integral into $tildey=frac1nu y$? It would then follow that
$$int_-infty^inftye^tildexiye^theta y-nu b(theta)f_0(y)dy=
int_-infty^inftye^xitildeye^nutheta tildey-nu b(theta)nuf_0(nutildey)dtildey=
int_-infty^inftye^xitildeye^nutheta tildey-nu b(theta)+ln(nuf_0(nutildey))dtildey.$$
The density would then be expressed as $f_hatY(y;theta, nu)=e^nu(theta tildey-b(theta))+ln(nuf_0(nutildey))$ and it would have the form in the desired form mentioned in the Edit above.
Would this make sense? My math background is not really so strong, so I have a feeling that I could have made some illegitimate operations here...
probability probability-theory moment-generating-functions
probability probability-theory moment-generating-functions
asked Apr 1 at 18:00
J.K.J.K.
1637
asked Apr 1 at 18:00
J.K.J.K.
1637
edited Apr 2 at 11:40
J.K.
asked Apr 1 at 18:00
J.K.J.K.
1637
asked Apr 1 at 18:00
J.K.J.K.
1637
asked Apr 1 at 18:00
J.K.J.K.
1637
1637
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
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%2f3170930%2ffinding-a-probability-density-from-an-exponential-family-using-a-moment-generati%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%2f3170930%2ffinding-a-probability-density-from-an-exponential-family-using-a-moment-generati%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
