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How can I understand the mean of convoluted Gaussian function is $mu_1 + mu_2$.



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)How can I make some progress on this Gaussian-looking integral?Integrating Normal Distribution with $1$, $x$, and $x^2$Functions with closed-form expectations under Gaussian PDFImpact of gaussian mean shift on top end distributionIntegral of the product of Gaussian PDF and Q function involving natural logGaussian Integrals and Grassmann AlgebraRelation between Taylor Expansion and multivariate Gaussian Distributionfinding area under the curve of a valueIs there an analytic form for the squared error of the difference of two univariate Gaussians?Kalman filter: the bayesian approach derivation some clarifications










0












$begingroup$


I struggle to understand the mean of convoluted two Gaussian functions is $mu_1 + mu_2$ instead of $(mu_1 + mu_2)/2$. Could someone provide visual explaination?










share|cite|improve this question











$endgroup$











  • $begingroup$
    Can you explain in your question why you think it should be halved? Do you understand that if $X$ and $Y$ have distributions $f$ and $g$ then $X + Y$ has distribution $f * g$?
    $endgroup$
    – Trevor Gunn
    Mar 27 at 20:13















0












$begingroup$


I struggle to understand the mean of convoluted two Gaussian functions is $mu_1 + mu_2$ instead of $(mu_1 + mu_2)/2$. Could someone provide visual explaination?










share|cite|improve this question











$endgroup$











  • $begingroup$
    Can you explain in your question why you think it should be halved? Do you understand that if $X$ and $Y$ have distributions $f$ and $g$ then $X + Y$ has distribution $f * g$?
    $endgroup$
    – Trevor Gunn
    Mar 27 at 20:13













0












0








0





$begingroup$


I struggle to understand the mean of convoluted two Gaussian functions is $mu_1 + mu_2$ instead of $(mu_1 + mu_2)/2$. Could someone provide visual explaination?










share|cite|improve this question











$endgroup$




I struggle to understand the mean of convoluted two Gaussian functions is $mu_1 + mu_2$ instead of $(mu_1 + mu_2)/2$. Could someone provide visual explaination?







gaussian-integral






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 27 at 20:03







Alex Gao

















asked Mar 27 at 19:57









Alex GaoAlex Gao

1125




1125











  • $begingroup$
    Can you explain in your question why you think it should be halved? Do you understand that if $X$ and $Y$ have distributions $f$ and $g$ then $X + Y$ has distribution $f * g$?
    $endgroup$
    – Trevor Gunn
    Mar 27 at 20:13
















  • $begingroup$
    Can you explain in your question why you think it should be halved? Do you understand that if $X$ and $Y$ have distributions $f$ and $g$ then $X + Y$ has distribution $f * g$?
    $endgroup$
    – Trevor Gunn
    Mar 27 at 20:13















$begingroup$
Can you explain in your question why you think it should be halved? Do you understand that if $X$ and $Y$ have distributions $f$ and $g$ then $X + Y$ has distribution $f * g$?
$endgroup$
– Trevor Gunn
Mar 27 at 20:13




$begingroup$
Can you explain in your question why you think it should be halved? Do you understand that if $X$ and $Y$ have distributions $f$ and $g$ then $X + Y$ has distribution $f * g$?
$endgroup$
– Trevor Gunn
Mar 27 at 20:13










1 Answer
1






active

oldest

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2












$begingroup$

If $X$ and $Y$ are two independent continuous random variables with means $E[X]=mu_X$ and $E[Y] = mu_Y$, then the density of their sum $X+Y$ is the convolution of the densities of $X$ and $Y$. In the special case when $X$ and $Y$ are Gaussian random variables, $X+Y$ is also a Gaussian random variable. But, regardless of whether $X$ and $Y$ are Gaussian or not, independent or not, or continuous or not, it is true that
$$E[X+Y] = E[X] + E[Y].$$
This result is referred to as the linearity of expectation, and explains why the mean of $X+Y$ is $mu_X + mu_Y$, and not the arithmetic average $fracmu_X+mu_Y2$ that you are hoping to see.



Visually, the densities of independent Gaussian $X$ and $Y$ are bell-shaped curves centered at $mu_X$ and $mu_Y$ respectively, and the value of the convolution of these densities at $z$ (a.k.a the numerical value of the density of $X+Y$ at $z$) is obtained by



  • "flipping" the density of $Y$ about the origin the result of which operation looks for all the world just like the density of $Y$ centered at $-mu_Y$ instead of at $mu_Y$.

  • sliding the flipped density to the right by $z$

  • multiplying the density of $X$ with the flipped and slid density of $Y$ and integrating the product from $-infty$ to $infty$.

If your brain has not already boggled at all these shenanigans and can still visualize a tad more, consider that the maximum value of the integral will occur when the two peaks align, that is, when the flipped density of $Y$ has been slid so that its peak at $-mu_Y$ has shifted due to the slide so that it coincides with the peak of the $X$ density which is at $mu_X$, i.e. a total slide of $mu_X+mu_Y$. So, the peak of the convolution is at $mu_X+mu_Y$ and so the mean of $X+Y$ is $mu_X+mu_Y$.






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    2












    $begingroup$

    If $X$ and $Y$ are two independent continuous random variables with means $E[X]=mu_X$ and $E[Y] = mu_Y$, then the density of their sum $X+Y$ is the convolution of the densities of $X$ and $Y$. In the special case when $X$ and $Y$ are Gaussian random variables, $X+Y$ is also a Gaussian random variable. But, regardless of whether $X$ and $Y$ are Gaussian or not, independent or not, or continuous or not, it is true that
    $$E[X+Y] = E[X] + E[Y].$$
    This result is referred to as the linearity of expectation, and explains why the mean of $X+Y$ is $mu_X + mu_Y$, and not the arithmetic average $fracmu_X+mu_Y2$ that you are hoping to see.



    Visually, the densities of independent Gaussian $X$ and $Y$ are bell-shaped curves centered at $mu_X$ and $mu_Y$ respectively, and the value of the convolution of these densities at $z$ (a.k.a the numerical value of the density of $X+Y$ at $z$) is obtained by



    • "flipping" the density of $Y$ about the origin the result of which operation looks for all the world just like the density of $Y$ centered at $-mu_Y$ instead of at $mu_Y$.

    • sliding the flipped density to the right by $z$

    • multiplying the density of $X$ with the flipped and slid density of $Y$ and integrating the product from $-infty$ to $infty$.

    If your brain has not already boggled at all these shenanigans and can still visualize a tad more, consider that the maximum value of the integral will occur when the two peaks align, that is, when the flipped density of $Y$ has been slid so that its peak at $-mu_Y$ has shifted due to the slide so that it coincides with the peak of the $X$ density which is at $mu_X$, i.e. a total slide of $mu_X+mu_Y$. So, the peak of the convolution is at $mu_X+mu_Y$ and so the mean of $X+Y$ is $mu_X+mu_Y$.






    share|cite|improve this answer











    $endgroup$

















      2












      $begingroup$

      If $X$ and $Y$ are two independent continuous random variables with means $E[X]=mu_X$ and $E[Y] = mu_Y$, then the density of their sum $X+Y$ is the convolution of the densities of $X$ and $Y$. In the special case when $X$ and $Y$ are Gaussian random variables, $X+Y$ is also a Gaussian random variable. But, regardless of whether $X$ and $Y$ are Gaussian or not, independent or not, or continuous or not, it is true that
      $$E[X+Y] = E[X] + E[Y].$$
      This result is referred to as the linearity of expectation, and explains why the mean of $X+Y$ is $mu_X + mu_Y$, and not the arithmetic average $fracmu_X+mu_Y2$ that you are hoping to see.



      Visually, the densities of independent Gaussian $X$ and $Y$ are bell-shaped curves centered at $mu_X$ and $mu_Y$ respectively, and the value of the convolution of these densities at $z$ (a.k.a the numerical value of the density of $X+Y$ at $z$) is obtained by



      • "flipping" the density of $Y$ about the origin the result of which operation looks for all the world just like the density of $Y$ centered at $-mu_Y$ instead of at $mu_Y$.

      • sliding the flipped density to the right by $z$

      • multiplying the density of $X$ with the flipped and slid density of $Y$ and integrating the product from $-infty$ to $infty$.

      If your brain has not already boggled at all these shenanigans and can still visualize a tad more, consider that the maximum value of the integral will occur when the two peaks align, that is, when the flipped density of $Y$ has been slid so that its peak at $-mu_Y$ has shifted due to the slide so that it coincides with the peak of the $X$ density which is at $mu_X$, i.e. a total slide of $mu_X+mu_Y$. So, the peak of the convolution is at $mu_X+mu_Y$ and so the mean of $X+Y$ is $mu_X+mu_Y$.






      share|cite|improve this answer











      $endgroup$















        2












        2








        2





        $begingroup$

        If $X$ and $Y$ are two independent continuous random variables with means $E[X]=mu_X$ and $E[Y] = mu_Y$, then the density of their sum $X+Y$ is the convolution of the densities of $X$ and $Y$. In the special case when $X$ and $Y$ are Gaussian random variables, $X+Y$ is also a Gaussian random variable. But, regardless of whether $X$ and $Y$ are Gaussian or not, independent or not, or continuous or not, it is true that
        $$E[X+Y] = E[X] + E[Y].$$
        This result is referred to as the linearity of expectation, and explains why the mean of $X+Y$ is $mu_X + mu_Y$, and not the arithmetic average $fracmu_X+mu_Y2$ that you are hoping to see.



        Visually, the densities of independent Gaussian $X$ and $Y$ are bell-shaped curves centered at $mu_X$ and $mu_Y$ respectively, and the value of the convolution of these densities at $z$ (a.k.a the numerical value of the density of $X+Y$ at $z$) is obtained by



        • "flipping" the density of $Y$ about the origin the result of which operation looks for all the world just like the density of $Y$ centered at $-mu_Y$ instead of at $mu_Y$.

        • sliding the flipped density to the right by $z$

        • multiplying the density of $X$ with the flipped and slid density of $Y$ and integrating the product from $-infty$ to $infty$.

        If your brain has not already boggled at all these shenanigans and can still visualize a tad more, consider that the maximum value of the integral will occur when the two peaks align, that is, when the flipped density of $Y$ has been slid so that its peak at $-mu_Y$ has shifted due to the slide so that it coincides with the peak of the $X$ density which is at $mu_X$, i.e. a total slide of $mu_X+mu_Y$. So, the peak of the convolution is at $mu_X+mu_Y$ and so the mean of $X+Y$ is $mu_X+mu_Y$.






        share|cite|improve this answer











        $endgroup$



        If $X$ and $Y$ are two independent continuous random variables with means $E[X]=mu_X$ and $E[Y] = mu_Y$, then the density of their sum $X+Y$ is the convolution of the densities of $X$ and $Y$. In the special case when $X$ and $Y$ are Gaussian random variables, $X+Y$ is also a Gaussian random variable. But, regardless of whether $X$ and $Y$ are Gaussian or not, independent or not, or continuous or not, it is true that
        $$E[X+Y] = E[X] + E[Y].$$
        This result is referred to as the linearity of expectation, and explains why the mean of $X+Y$ is $mu_X + mu_Y$, and not the arithmetic average $fracmu_X+mu_Y2$ that you are hoping to see.



        Visually, the densities of independent Gaussian $X$ and $Y$ are bell-shaped curves centered at $mu_X$ and $mu_Y$ respectively, and the value of the convolution of these densities at $z$ (a.k.a the numerical value of the density of $X+Y$ at $z$) is obtained by



        • "flipping" the density of $Y$ about the origin the result of which operation looks for all the world just like the density of $Y$ centered at $-mu_Y$ instead of at $mu_Y$.

        • sliding the flipped density to the right by $z$

        • multiplying the density of $X$ with the flipped and slid density of $Y$ and integrating the product from $-infty$ to $infty$.

        If your brain has not already boggled at all these shenanigans and can still visualize a tad more, consider that the maximum value of the integral will occur when the two peaks align, that is, when the flipped density of $Y$ has been slid so that its peak at $-mu_Y$ has shifted due to the slide so that it coincides with the peak of the $X$ density which is at $mu_X$, i.e. a total slide of $mu_X+mu_Y$. So, the peak of the convolution is at $mu_X+mu_Y$ and so the mean of $X+Y$ is $mu_X+mu_Y$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Mar 27 at 20:33

























        answered Mar 27 at 20:16









        Dilip SarwateDilip Sarwate

        19.5k13078




        19.5k13078



























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