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
How do I find out the mythology and history of my Fortress?
Do wooden building fires get hotter than 600°C?
AppleTVs create a chatty alternate WiFi network
Drawing without replacement: why is the order of draw irrelevant?
How would a mousetrap for use in space work?
If Windows 7 doesn't support WSL, then what does Linux subsystem option mean?
Crossing US/Canada Border for less than 24 hours
Why does it sometimes sound good to play a grace note as a lead in to a note in a melody?
Dating a Former Employee
What's the meaning of "fortified infraction restraint"?
Disembodied hand growing fangs
Maximum summed subsequences with non-adjacent items
What are the out-of-universe reasons for the references to Toby Maguire-era Spider-Man in Into the Spider-Verse?
Is a ledger board required if the side of my house is wood?
Did Deadpool rescue all of the X-Force?
How could we fake a moon landing now?
Why should I vote and accept answers?
How to write this math term? with cases it isn't working
Why weren't discrete x86 CPUs ever used in game hardware?
SF book about people trapped in a series of worlds they imagine
Is CEO the "profession" with the most psychopaths?
Is there hard evidence that the grant peer review system performs significantly better than random?
Time to Settle Down!
Chinese Seal on silk painting - what does it mean?
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
