Showing $ mathbb{P}Big( frac{Pi - lambda}{sqrt{lambda}}leq x Big) $ [duplicate]

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  Proof of $frac{Y^{lambda}-lambda}{sqrt{lambda}}to Zsim N(0,1)$ in distribution as $lambdatoinfty$?
  
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Let $Pi$ be a random variable distributed by Poisson distribution with parameter $lambda>0.$ Need to show that $$ mathbb{P}Big( frac{Pi - lambda}{sqrt{lambda}}leq x Big) rightarrow_{lambda to infty} Phi(x)$$ for every $x in mathbb{R}.$

I have no idea what to do and how to start. I know that Poisson random variable $exp(iat + lambda(e^{ibt-1})$ maybe this is equal to $Phi(t)$ but than what is $a$ and $b$?

probability probability-theory poisson-distribution characteristic-functions

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asked Jan 24 at 13:34

AtstovasAtstovas

1139

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marked as duplicate by NCh, Misha Lavrov, max_zorn, Cesareo, Shailesh Jan 27 at 3:31

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Central limit theorem is an answer for your question.
  $endgroup$
  – vermator
  Jan 24 at 13:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Look the answer here: math.stackexchange.com/questions/245379/…
  $endgroup$
  – user52227
  Jan 24 at 20:01
  

  
  
  
  

add a comment | 

-1

$begingroup$

This question already has an answer here:

• 
  Proof of $frac{Y^{lambda}-lambda}{sqrt{lambda}}to Zsim N(0,1)$ in distribution as $lambdatoinfty$?
  
  1 answer
  
  

Let $Pi$ be a random variable distributed by Poisson distribution with parameter $lambda>0.$ Need to show that $$ mathbb{P}Big( frac{Pi - lambda}{sqrt{lambda}}leq x Big) rightarrow_{lambda to infty} Phi(x)$$ for every $x in mathbb{R}.$

I have no idea what to do and how to start. I know that Poisson random variable $exp(iat + lambda(e^{ibt-1})$ maybe this is equal to $Phi(t)$ but than what is $a$ and $b$?

probability probability-theory poisson-distribution characteristic-functions

share|cite|improve this question

asked Jan 24 at 13:34

AtstovasAtstovas

1139

$endgroup$

marked as duplicate by NCh, Misha Lavrov, max_zorn, Cesareo, Shailesh Jan 27 at 3:31

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Central limit theorem is an answer for your question.
  $endgroup$
  – vermator
  Jan 24 at 13:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Look the answer here: math.stackexchange.com/questions/245379/…
  $endgroup$
  – user52227
  Jan 24 at 20:01
  

  
  
  
  

add a comment | 

-1

-1

-1

$begingroup$

This question already has an answer here:

• 
  Proof of $frac{Y^{lambda}-lambda}{sqrt{lambda}}to Zsim N(0,1)$ in distribution as $lambdatoinfty$?
  
  1 answer
  
  

Let $Pi$ be a random variable distributed by Poisson distribution with parameter $lambda>0.$ Need to show that $$ mathbb{P}Big( frac{Pi - lambda}{sqrt{lambda}}leq x Big) rightarrow_{lambda to infty} Phi(x)$$ for every $x in mathbb{R}.$

I have no idea what to do and how to start. I know that Poisson random variable $exp(iat + lambda(e^{ibt-1})$ maybe this is equal to $Phi(t)$ but than what is $a$ and $b$?

probability probability-theory poisson-distribution characteristic-functions

share|cite|improve this question

asked Jan 24 at 13:34

AtstovasAtstovas

1139

$endgroup$

This question already has an answer here:

• 
  Proof of $frac{Y^{lambda}-lambda}{sqrt{lambda}}to Zsim N(0,1)$ in distribution as $lambdatoinfty$?
  
  1 answer
  
  

Let $Pi$ be a random variable distributed by Poisson distribution with parameter $lambda>0.$ Need to show that $$ mathbb{P}Big( frac{Pi - lambda}{sqrt{lambda}}leq x Big) rightarrow_{lambda to infty} Phi(x)$$ for every $x in mathbb{R}.$

I have no idea what to do and how to start. I know that Poisson random variable $exp(iat + lambda(e^{ibt-1})$ maybe this is equal to $Phi(t)$ but than what is $a$ and $b$?

This question already has an answer here:

• 
  Proof of $frac{Y^{lambda}-lambda}{sqrt{lambda}}to Zsim N(0,1)$ in distribution as $lambdatoinfty$?
  
  1 answer
  
  

probability probability-theory poisson-distribution characteristic-functions

probability probability-theory poisson-distribution characteristic-functions

share|cite|improve this question

asked Jan 24 at 13:34

AtstovasAtstovas

1139

share|cite|improve this question

asked Jan 24 at 13:34

AtstovasAtstovas

1139

share|cite|improve this question

share|cite|improve this question

asked Jan 24 at 13:34

AtstovasAtstovas

1139

asked Jan 24 at 13:34

AtstovasAtstovas

1139

asked Jan 24 at 13:34

AtstovasAtstovas

1139

1139

marked as duplicate by NCh, Misha Lavrov, max_zorn, Cesareo, Shailesh Jan 27 at 3:31

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

marked as duplicate by NCh, Misha Lavrov, max_zorn, Cesareo, Shailesh Jan 27 at 3:31

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Central limit theorem is an answer for your question.
  $endgroup$
  – vermator
  Jan 24 at 13:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Look the answer here: math.stackexchange.com/questions/245379/…
  $endgroup$
  – user52227
  Jan 24 at 20:01
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Central limit theorem is an answer for your question.
  $endgroup$
  – vermator
  Jan 24 at 13:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Look the answer here: math.stackexchange.com/questions/245379/…
  $endgroup$
  – user52227
  Jan 24 at 20:01
  

  
  
  
  

$begingroup$
Central limit theorem is an answer for your question.
$endgroup$
– vermator
Jan 24 at 13:37

$begingroup$
Central limit theorem is an answer for your question.
$endgroup$
– vermator
Jan 24 at 13:37

$begingroup$
Look the answer here: math.stackexchange.com/questions/245379/…
$endgroup$
– user52227
Jan 24 at 20:01

$begingroup$
Look the answer here: math.stackexchange.com/questions/245379/…
$endgroup$
– user52227
Jan 24 at 20:01

add a comment | 

1 Answer
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Hint: Use the central limit theorem and the fact that exponentials transform sums in products.

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answered Jan 24 at 13:36

Math_QEDMath_QED

7,71031453

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

0

$begingroup$

Hint: Use the central limit theorem and the fact that exponentials transform sums in products.

share|cite|improve this answer

answered Jan 24 at 13:36

Math_QEDMath_QED

7,71031453

$endgroup$

add a comment | 

0

$begingroup$

Hint: Use the central limit theorem and the fact that exponentials transform sums in products.

share|cite|improve this answer

answered Jan 24 at 13:36

Math_QEDMath_QED

7,71031453

$endgroup$

add a comment | 

0

0

0

$begingroup$

Hint: Use the central limit theorem and the fact that exponentials transform sums in products.

share|cite|improve this answer

answered Jan 24 at 13:36

Math_QEDMath_QED

7,71031453

$endgroup$

Hint: Use the central limit theorem and the fact that exponentials transform sums in products.

share|cite|improve this answer

answered Jan 24 at 13:36

Math_QEDMath_QED

7,71031453

share|cite|improve this answer

share|cite|improve this answer

answered Jan 24 at 13:36

Math_QEDMath_QED

7,71031453

answered Jan 24 at 13:36

Math_QEDMath_QED

7,71031453

answered Jan 24 at 13:36

Math_QEDMath_QED

7,71031453

7,71031453

add a comment | 

add a comment | 

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