Two stability of normal random variable detailed explanation












0












$begingroup$


I am reading about two stability of normal random variable and I am super confused (no good background in stats or calc), but I really need to understand it. Please, see the following:
enter image description here



I first got stuck on "By definition, $f_z(z) = int f_x(z-y) f_y(y) dy$"



Could someone explain me how we get $f_z$ ? I do not really understand the intuition behind the integral. I am further confused about the computation. It would be great if you could show me intermediate steps of this computation (integral) so I can follow. I will really appreciate!










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  • $begingroup$
    It is a convolution (the keyword might help): summing independent random variables corresponds to doing a convolution of their probability density functions.
    $endgroup$
    – Clement C.
    Dec 16 '18 at 2:03






  • 1




    $begingroup$
    I'm a little disturbed that the "claim" fails to mention that $X$ and $Y$ are independent.
    $endgroup$
    – zoidberg
    Dec 16 '18 at 2:48










  • $begingroup$
    @ClementC. thanks for the reference!
    $endgroup$
    – YohanRoth
    Dec 16 '18 at 4:21
















0












$begingroup$


I am reading about two stability of normal random variable and I am super confused (no good background in stats or calc), but I really need to understand it. Please, see the following:
enter image description here



I first got stuck on "By definition, $f_z(z) = int f_x(z-y) f_y(y) dy$"



Could someone explain me how we get $f_z$ ? I do not really understand the intuition behind the integral. I am further confused about the computation. It would be great if you could show me intermediate steps of this computation (integral) so I can follow. I will really appreciate!










share|cite|improve this question









$endgroup$












  • $begingroup$
    It is a convolution (the keyword might help): summing independent random variables corresponds to doing a convolution of their probability density functions.
    $endgroup$
    – Clement C.
    Dec 16 '18 at 2:03






  • 1




    $begingroup$
    I'm a little disturbed that the "claim" fails to mention that $X$ and $Y$ are independent.
    $endgroup$
    – zoidberg
    Dec 16 '18 at 2:48










  • $begingroup$
    @ClementC. thanks for the reference!
    $endgroup$
    – YohanRoth
    Dec 16 '18 at 4:21














0












0








0





$begingroup$


I am reading about two stability of normal random variable and I am super confused (no good background in stats or calc), but I really need to understand it. Please, see the following:
enter image description here



I first got stuck on "By definition, $f_z(z) = int f_x(z-y) f_y(y) dy$"



Could someone explain me how we get $f_z$ ? I do not really understand the intuition behind the integral. I am further confused about the computation. It would be great if you could show me intermediate steps of this computation (integral) so I can follow. I will really appreciate!










share|cite|improve this question









$endgroup$




I am reading about two stability of normal random variable and I am super confused (no good background in stats or calc), but I really need to understand it. Please, see the following:
enter image description here



I first got stuck on "By definition, $f_z(z) = int f_x(z-y) f_y(y) dy$"



Could someone explain me how we get $f_z$ ? I do not really understand the intuition behind the integral. I am further confused about the computation. It would be great if you could show me intermediate steps of this computation (integral) so I can follow. I will really appreciate!







calculus probability statistics






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 16 '18 at 1:54









YohanRothYohanRoth

6381714




6381714












  • $begingroup$
    It is a convolution (the keyword might help): summing independent random variables corresponds to doing a convolution of their probability density functions.
    $endgroup$
    – Clement C.
    Dec 16 '18 at 2:03






  • 1




    $begingroup$
    I'm a little disturbed that the "claim" fails to mention that $X$ and $Y$ are independent.
    $endgroup$
    – zoidberg
    Dec 16 '18 at 2:48










  • $begingroup$
    @ClementC. thanks for the reference!
    $endgroup$
    – YohanRoth
    Dec 16 '18 at 4:21


















  • $begingroup$
    It is a convolution (the keyword might help): summing independent random variables corresponds to doing a convolution of their probability density functions.
    $endgroup$
    – Clement C.
    Dec 16 '18 at 2:03






  • 1




    $begingroup$
    I'm a little disturbed that the "claim" fails to mention that $X$ and $Y$ are independent.
    $endgroup$
    – zoidberg
    Dec 16 '18 at 2:48










  • $begingroup$
    @ClementC. thanks for the reference!
    $endgroup$
    – YohanRoth
    Dec 16 '18 at 4:21
















$begingroup$
It is a convolution (the keyword might help): summing independent random variables corresponds to doing a convolution of their probability density functions.
$endgroup$
– Clement C.
Dec 16 '18 at 2:03




$begingroup$
It is a convolution (the keyword might help): summing independent random variables corresponds to doing a convolution of their probability density functions.
$endgroup$
– Clement C.
Dec 16 '18 at 2:03




1




1




$begingroup$
I'm a little disturbed that the "claim" fails to mention that $X$ and $Y$ are independent.
$endgroup$
– zoidberg
Dec 16 '18 at 2:48




$begingroup$
I'm a little disturbed that the "claim" fails to mention that $X$ and $Y$ are independent.
$endgroup$
– zoidberg
Dec 16 '18 at 2:48












$begingroup$
@ClementC. thanks for the reference!
$endgroup$
– YohanRoth
Dec 16 '18 at 4:21




$begingroup$
@ClementC. thanks for the reference!
$endgroup$
– YohanRoth
Dec 16 '18 at 4:21










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