Prove that the given RV is normal
$begingroup$
This is an old qualifying exam problem of probability theory.
Let $X,Y$ be two iid random variables with mean zero and variance 1. Suppose that $X,Y,frac{X+Y}{sqrt{2}}$ are all identically distributed. Prove that $Xsim N(0,1)$.
My attempt: Let $varphi$ be the characteristic function of $X$. Then by the given condition, I got $varphi(t)=varphi(t/sqrt{2})^2$. Then I'm trying to conclude $varphi(t)=e^{-t^2/2}$ from such functional equation, but it seems not work.
Does anyone have ideas?
Thanks in advance!
probability-theory probability-distributions characteristic-functions
edited Dec 6 '18 at 14:06
gt6989b
33.6k22453
asked Dec 6 '18 at 13:55
bellcirclebellcircle
1,331411
$endgroup$
add a comment |
$begingroup$
This is an old qualifying exam problem of probability theory.
Let $X,Y$ be two iid random variables with mean zero and variance 1. Suppose that $X,Y,frac{X+Y}{sqrt{2}}$ are all identically distributed. Prove that $Xsim N(0,1)$.
My attempt: Let $varphi$ be the characteristic function of $X$. Then by the given condition, I got $varphi(t)=varphi(t/sqrt{2})^2$. Then I'm trying to conclude $varphi(t)=e^{-t^2/2}$ from such functional equation, but it seems not work.
Does anyone have ideas?
Thanks in advance!
probability-theory probability-distributions characteristic-functions
edited Dec 6 '18 at 14:06
gt6989b
33.6k22453
asked Dec 6 '18 at 13:55
bellcirclebellcircle
1,331411
$endgroup$
2
$begingroup$
Hint: Iterate the relation you found and use the expansion $varphi(t)=1-t^2/2+o(t^2)$ when $tto0$.
$endgroup$
– Did
Dec 6 '18 at 14:10
$begingroup$
@Did Thanks for your hint! I'll post my answer shortly.
$endgroup$
– bellcircle
Dec 6 '18 at 15:38
add a comment |
$begingroup$
This is an old qualifying exam problem of probability theory.
Let $X,Y$ be two iid random variables with mean zero and variance 1. Suppose that $X,Y,frac{X+Y}{sqrt{2}}$ are all identically distributed. Prove that $Xsim N(0,1)$.
My attempt: Let $varphi$ be the characteristic function of $X$. Then by the given condition, I got $varphi(t)=varphi(t/sqrt{2})^2$. Then I'm trying to conclude $varphi(t)=e^{-t^2/2}$ from such functional equation, but it seems not work.
Does anyone have ideas?
Thanks in advance!
probability-theory probability-distributions characteristic-functions
edited Dec 6 '18 at 14:06
gt6989b
33.6k22453
asked Dec 6 '18 at 13:55
bellcirclebellcircle
1,331411
$endgroup$
This is an old qualifying exam problem of probability theory.
Let $X,Y$ be two iid random variables with mean zero and variance 1. Suppose that $X,Y,frac{X+Y}{sqrt{2}}$ are all identically distributed. Prove that $Xsim N(0,1)$.
My attempt: Let $varphi$ be the characteristic function of $X$. Then by the given condition, I got $varphi(t)=varphi(t/sqrt{2})^2$. Then I'm trying to conclude $varphi(t)=e^{-t^2/2}$ from such functional equation, but it seems not work.
Does anyone have ideas?
Thanks in advance!
probability-theory probability-distributions characteristic-functions
probability-theory probability-distributions characteristic-functions
edited Dec 6 '18 at 14:06
gt6989b
33.6k22453
asked Dec 6 '18 at 13:55
bellcirclebellcircle
1,331411
edited Dec 6 '18 at 14:06
gt6989b
33.6k22453
asked Dec 6 '18 at 13:55
bellcirclebellcircle
1,331411
edited Dec 6 '18 at 14:06
gt6989b
33.6k22453
edited Dec 6 '18 at 14:06
gt6989b
33.6k22453
edited Dec 6 '18 at 14:06
gt6989b
33.6k22453
33.6k22453
asked Dec 6 '18 at 13:55
bellcirclebellcircle
1,331411
asked Dec 6 '18 at 13:55
bellcirclebellcircle
1,331411
asked Dec 6 '18 at 13:55
bellcirclebellcircle
1,331411
1,331411
2
$begingroup$
Hint: Iterate the relation you found and use the expansion $varphi(t)=1-t^2/2+o(t^2)$ when $tto0$.
$endgroup$
– Did
Dec 6 '18 at 14:10
$begingroup$
@Did Thanks for your hint! I'll post my answer shortly.
$endgroup$
– bellcircle
Dec 6 '18 at 15:38
add a comment |
2
$begingroup$
Hint: Iterate the relation you found and use the expansion $varphi(t)=1-t^2/2+o(t^2)$ when $tto0$.
$endgroup$
– Did
Dec 6 '18 at 14:10
$begingroup$
@Did Thanks for your hint! I'll post my answer shortly.
$endgroup$
– bellcircle
Dec 6 '18 at 15:38
2
2
$begingroup$
Hint: Iterate the relation you found and use the expansion $varphi(t)=1-t^2/2+o(t^2)$ when $tto0$.
$endgroup$
– Did
Dec 6 '18 at 14:10
$begingroup$
Hint: Iterate the relation you found and use the expansion $varphi(t)=1-t^2/2+o(t^2)$ when $tto0$.
$endgroup$
– Did
Dec 6 '18 at 14:10
$begingroup$
@Did Thanks for your hint! I'll post my answer shortly.
$endgroup$
– bellcircle
Dec 6 '18 at 15:38
$begingroup$
@Did Thanks for your hint! I'll post my answer shortly.
$endgroup$
– bellcircle
Dec 6 '18 at 15:38
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Iterating the property of $varphi$ found in the question, one gets $varphi(t)=left(varphi(t/2^{n/2})right)^{2^n}$ for all $ninmathbb{N}$.
Also, since $X$ has mean zero and variance 1, $varphi(t)=1-frac{t^2}{2}+o(t^2)$ as $tto 0$.
Therefore, $$varphi(t)=left(1-frac{t^2}{2^{n+1}}+o(frac{t^2}{2^n})right)^{2^n}=lim_{ntoinfty}left(1-frac{t^2}{2^{n+1}}+o(frac{t^2}{2^n})right)^{2^n}=e^{-t^2/2} $$ as desired.
answered Dec 6 '18 at 15:41
bellcirclebellcircle
1,331411
$endgroup$
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Dec 6 '18 at 16:03
add a comment |
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1 Answer
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active
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1 Answer
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active
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$begingroup$
Iterating the property of $varphi$ found in the question, one gets $varphi(t)=left(varphi(t/2^{n/2})right)^{2^n}$ for all $ninmathbb{N}$.
Also, since $X$ has mean zero and variance 1, $varphi(t)=1-frac{t^2}{2}+o(t^2)$ as $tto 0$.
Therefore, $$varphi(t)=left(1-frac{t^2}{2^{n+1}}+o(frac{t^2}{2^n})right)^{2^n}=lim_{ntoinfty}left(1-frac{t^2}{2^{n+1}}+o(frac{t^2}{2^n})right)^{2^n}=e^{-t^2/2} $$ as desired.
answered Dec 6 '18 at 15:41
bellcirclebellcircle
1,331411
$endgroup$
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Dec 6 '18 at 16:03
add a comment |
$begingroup$
Iterating the property of $varphi$ found in the question, one gets $varphi(t)=left(varphi(t/2^{n/2})right)^{2^n}$ for all $ninmathbb{N}$.
Also, since $X$ has mean zero and variance 1, $varphi(t)=1-frac{t^2}{2}+o(t^2)$ as $tto 0$.
Therefore, $$varphi(t)=left(1-frac{t^2}{2^{n+1}}+o(frac{t^2}{2^n})right)^{2^n}=lim_{ntoinfty}left(1-frac{t^2}{2^{n+1}}+o(frac{t^2}{2^n})right)^{2^n}=e^{-t^2/2} $$ as desired.
answered Dec 6 '18 at 15:41
bellcirclebellcircle
1,331411
$endgroup$
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Dec 6 '18 at 16:03
add a comment |
$begingroup$
Iterating the property of $varphi$ found in the question, one gets $varphi(t)=left(varphi(t/2^{n/2})right)^{2^n}$ for all $ninmathbb{N}$.
Also, since $X$ has mean zero and variance 1, $varphi(t)=1-frac{t^2}{2}+o(t^2)$ as $tto 0$.
Therefore, $$varphi(t)=left(1-frac{t^2}{2^{n+1}}+o(frac{t^2}{2^n})right)^{2^n}=lim_{ntoinfty}left(1-frac{t^2}{2^{n+1}}+o(frac{t^2}{2^n})right)^{2^n}=e^{-t^2/2} $$ as desired.
answered Dec 6 '18 at 15:41
bellcirclebellcircle
1,331411
$endgroup$
Iterating the property of $varphi$ found in the question, one gets $varphi(t)=left(varphi(t/2^{n/2})right)^{2^n}$ for all $ninmathbb{N}$.
Also, since $X$ has mean zero and variance 1, $varphi(t)=1-frac{t^2}{2}+o(t^2)$ as $tto 0$.
Therefore, $$varphi(t)=left(1-frac{t^2}{2^{n+1}}+o(frac{t^2}{2^n})right)^{2^n}=lim_{ntoinfty}left(1-frac{t^2}{2^{n+1}}+o(frac{t^2}{2^n})right)^{2^n}=e^{-t^2/2} $$ as desired.
answered Dec 6 '18 at 15:41
bellcirclebellcircle
1,331411
answered Dec 6 '18 at 15:41
bellcirclebellcircle
1,331411
answered Dec 6 '18 at 15:41
bellcirclebellcircle
1,331411
answered Dec 6 '18 at 15:41
bellcirclebellcircle
1,331411
1,331411
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Dec 6 '18 at 16:03
add a comment |
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Dec 6 '18 at 16:03
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Dec 6 '18 at 16:03
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Dec 6 '18 at 16:03
add a comment |
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$begingroup$
Hint: Iterate the relation you found and use the expansion $varphi(t)=1-t^2/2+o(t^2)$ when $tto0$.
$endgroup$
– Did
Dec 6 '18 at 14:10
$begingroup$
@Did Thanks for your hint! I'll post my answer shortly.
$endgroup$
– bellcircle
Dec 6 '18 at 15:38