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 $P_M(M_B(t))$
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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
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oldest
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$begingroup$
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