Sum of the Poisson distribution (my solution vs. textbook) The Next CEO of Stack OverflowHow to find probability distribution function given the Moment Generating FunctionMoment generating function of a compound Poisson processMoment generating function within another functionA Poisson distribution problemMoment generating function of sum of $N$ exponentially distributed random variablesMarginal distribution of sum of Poisson random variables from i=2 to nApproximate values of $operatornameE[sqrt X]$ and $operatornameVar[sqrt X]$ for $X$ Poisson distributed with parameter $lambdatoinfty$Moment generation function - compound PoissonFind the distribution of $Z=X+Y$ where both $X$ and $Y$ are exponentially distributed.What is the distribution of $P_M(M_B(t))$
Can a single photon have an energy density?
How do spells that require an ability check vs. the caster's spell save DC work?
Implement the Thanos sorting algorithm
Go Pregnant or Go Home
How do I solve this limit?
Is it safe to use c_str() on a temporary string?
Whats the best way to handle refactoring a big file?
Which organization defines CJK Unified Ideographs?
Apart from "berlinern", do any other German dialects have a corresponding verb?
Why is there a PLL in CPU?
Too much space between section and text in a twocolumn document
Inappropriate reference requests from Journal reviewers
What's the point of interval inversion?
Only print output after finding pattern
Trouble understanding the speech of overseas colleagues
How to count occurrences of text in a file?
Can I equip Skullclamp on a creature I am sacrificing?
How to use tikz in fbox?
Failed to fetch jessie backports repository
What can we do to stop prior company from asking us questions?
The King's new dress
How can I quit an app using Terminal?
What does this shorthand mean?
Return the Closest Prime Number
Sum of the Poisson distribution (my solution vs. textbook)
The Next CEO of Stack OverflowHow to find probability distribution function given the Moment Generating FunctionMoment generating function of a compound Poisson processMoment generating function within another functionA Poisson distribution problemMoment generating function of sum of $N$ exponentially distributed random variablesMarginal distribution of sum of Poisson random variables from i=2 to nApproximate values of $operatornameE[sqrt X]$ and $operatornameVar[sqrt X]$ for $X$ Poisson distributed with parameter $lambdatoinfty$Moment generation function - compound PoissonFind the distribution of $Z=X+Y$ where both $X$ and $Y$ are exponentially distributed.What is the distribution of $P_M(M_B(t))$
$begingroup$
I feel something is wrong, but can't place it:
Assume $X_i$ are i.i.d. Poisson distribution with parameter $lambda$
and define
$$Y = sum_i=1^n X_i $$
$$M_X(t) = exp((e^t-1)cdotlambda)$$
$$M_Y(t) = Big(exp((e^t-1)cdotlambda)Big)^n
= exp((e^t-1)cdot ncdotlambda)$$
where $M_X(cdot)$ is moment generating function.
so I think $$f_Y(y)=textThe pdf of Y = e^-nlambda(nlambda)^yover y!$$
but in my textbook $$f_Y(y) = e^-nlambda(lambda)^yover y!$$
I don't understand it!
Is there something wrong in my solution?
or
is the textbook wrong?
summation poisson-distribution moment-generating-functions
edited Mar 18 at 12:47
Mostafa Ayaz
18.1k31040
asked Mar 18 at 11:46
JeonSeokJunJeonSeokJun
33
$endgroup$
add a comment |
$begingroup$
I feel something is wrong, but can't place it:
Assume $X_i$ are i.i.d. Poisson distribution with parameter $lambda$
and define
$$Y = sum_i=1^n X_i $$
$$M_X(t) = exp((e^t-1)cdotlambda)$$
$$M_Y(t) = Big(exp((e^t-1)cdotlambda)Big)^n
= exp((e^t-1)cdot ncdotlambda)$$
where $M_X(cdot)$ is moment generating function.
so I think $$f_Y(y)=textThe pdf of Y = e^-nlambda(nlambda)^yover y!$$
but in my textbook $$f_Y(y) = e^-nlambda(lambda)^yover y!$$
I don't understand it!
Is there something wrong in my solution?
or
is the textbook wrong?
summation poisson-distribution moment-generating-functions
edited Mar 18 at 12:47
Mostafa Ayaz
18.1k31040
asked Mar 18 at 11:46
JeonSeokJunJeonSeokJun
33
$endgroup$
1
$begingroup$
Hi, welcome to MSE. The site uses MathJax for displaying mathematical formulas. You should visit math.meta.stackexchange.com/questions/5020/… to format your question. It will be far easier to read, and you'll get more answers.
$endgroup$
– Thomas Lesgourgues
Mar 18 at 11:56
$begingroup$
What is $f(y)$?
$endgroup$
– Kavi Rama Murthy
Mar 18 at 12:13
add a comment |
$begingroup$
I feel something is wrong, but can't place it:
Assume $X_i$ are i.i.d. Poisson distribution with parameter $lambda$
and define
$$Y = sum_i=1^n X_i $$
$$M_X(t) = exp((e^t-1)cdotlambda)$$
$$M_Y(t) = Big(exp((e^t-1)cdotlambda)Big)^n
= exp((e^t-1)cdot ncdotlambda)$$
where $M_X(cdot)$ is moment generating function.
so I think $$f_Y(y)=textThe pdf of Y = e^-nlambda(nlambda)^yover y!$$
but in my textbook $$f_Y(y) = e^-nlambda(lambda)^yover y!$$
I don't understand it!
Is there something wrong in my solution?
or
is the textbook wrong?
summation poisson-distribution moment-generating-functions
edited Mar 18 at 12:47
Mostafa Ayaz
18.1k31040
asked Mar 18 at 11:46
JeonSeokJunJeonSeokJun
33
$endgroup$
I feel something is wrong, but can't place it:
Assume $X_i$ are i.i.d. Poisson distribution with parameter $lambda$
and define
$$Y = sum_i=1^n X_i $$
$$M_X(t) = exp((e^t-1)cdotlambda)$$
$$M_Y(t) = Big(exp((e^t-1)cdotlambda)Big)^n
= exp((e^t-1)cdot ncdotlambda)$$
where $M_X(cdot)$ is moment generating function.
so I think $$f_Y(y)=textThe pdf of Y = e^-nlambda(nlambda)^yover y!$$
but in my textbook $$f_Y(y) = e^-nlambda(lambda)^yover y!$$
I don't understand it!
Is there something wrong in my solution?
or
is the textbook wrong?
summation poisson-distribution moment-generating-functions
summation poisson-distribution moment-generating-functions
edited Mar 18 at 12:47
Mostafa Ayaz
18.1k31040
asked Mar 18 at 11:46
JeonSeokJunJeonSeokJun
33
edited Mar 18 at 12:47
Mostafa Ayaz
18.1k31040
asked Mar 18 at 11:46
JeonSeokJunJeonSeokJun
33
edited Mar 18 at 12:47
Mostafa Ayaz
18.1k31040
edited Mar 18 at 12:47
Mostafa Ayaz
18.1k31040
edited Mar 18 at 12:47
Mostafa Ayaz
18.1k31040
18.1k31040
asked Mar 18 at 11:46
JeonSeokJunJeonSeokJun
33
asked Mar 18 at 11:46
JeonSeokJunJeonSeokJun
33
asked Mar 18 at 11:46
JeonSeokJunJeonSeokJun
33
33
1
$begingroup$
Hi, welcome to MSE. The site uses MathJax for displaying mathematical formulas. You should visit math.meta.stackexchange.com/questions/5020/… to format your question. It will be far easier to read, and you'll get more answers.
$endgroup$
– Thomas Lesgourgues
Mar 18 at 11:56
$begingroup$
What is $f(y)$?
$endgroup$
– Kavi Rama Murthy
Mar 18 at 12:13
add a comment |
1
$begingroup$
Hi, welcome to MSE. The site uses MathJax for displaying mathematical formulas. You should visit math.meta.stackexchange.com/questions/5020/… to format your question. It will be far easier to read, and you'll get more answers.
$endgroup$
– Thomas Lesgourgues
Mar 18 at 11:56
$begingroup$
What is $f(y)$?
$endgroup$
– Kavi Rama Murthy
Mar 18 at 12:13
1
1
$begingroup$
Hi, welcome to MSE. The site uses MathJax for displaying mathematical formulas. You should visit math.meta.stackexchange.com/questions/5020/… to format your question. It will be far easier to read, and you'll get more answers.
$endgroup$
– Thomas Lesgourgues
Mar 18 at 11:56
$begingroup$
Hi, welcome to MSE. The site uses MathJax for displaying mathematical formulas. You should visit math.meta.stackexchange.com/questions/5020/… to format your question. It will be far easier to read, and you'll get more answers.
$endgroup$
– Thomas Lesgourgues
Mar 18 at 11:56
$begingroup$
What is $f(y)$?
$endgroup$
– Kavi Rama Murthy
Mar 18 at 12:13
$begingroup$
What is $f(y)$?
$endgroup$
– Kavi Rama Murthy
Mar 18 at 12:13
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%2f3152694%2fsum-of-the-poisson-distribution-my-solution-vs-textbook%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%2f3152694%2fsum-of-the-poisson-distribution-my-solution-vs-textbook%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
$begingroup$
Hi, welcome to MSE. The site uses MathJax for displaying mathematical formulas. You should visit math.meta.stackexchange.com/questions/5020/… to format your question. It will be far easier to read, and you'll get more answers.
$endgroup$
– Thomas Lesgourgues
Mar 18 at 11:56
$begingroup$
What is $f(y)$?
$endgroup$
– Kavi Rama Murthy
Mar 18 at 12:13