The expectation of absolute value of the sum of n i.i.d. random variables












1












$begingroup$


Let $varphi_i$ be a Gaussian random variable such that
$$varphi_i sim N(0,sigma^2), quad i = 1,2,ldots,n.$$
What's the expectation:
$$Eleft(left | sum_{i=1}^n e^{j varphi_i} right |right) $$
where $left | cdot right |$ is the absolute value operation and $j = sqrt{-1}$.






probability statistics absolute-value expected-value







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  • $begingroup$
    I don't know if there is a nice closed form for the absolute value but for the square of the absolute value the expectation is $frac{n (n-1)}{exp left(sigma ^2right)}+n$.
    $endgroup$
    – JimB
    Jan 1 at 19:55
















1












$begingroup$


Let $varphi_i$ be a Gaussian random variable such that
$$varphi_i sim N(0,sigma^2), quad i = 1,2,ldots,n.$$
What's the expectation:
$$Eleft(left | sum_{i=1}^n e^{j varphi_i} right |right) $$
where $left | cdot right |$ is the absolute value operation and $j = sqrt{-1}$.






probability statistics absolute-value expected-value







share|cite|improve this question









$endgroup$












  • $begingroup$
    I don't know if there is a nice closed form for the absolute value but for the square of the absolute value the expectation is $frac{n (n-1)}{exp left(sigma ^2right)}+n$.
    $endgroup$
    – JimB
    Jan 1 at 19:55














1












1








1


0



$begingroup$


Let $varphi_i$ be a Gaussian random variable such that
$$varphi_i sim N(0,sigma^2), quad i = 1,2,ldots,n.$$
What's the expectation:
$$Eleft(left | sum_{i=1}^n e^{j varphi_i} right |right) $$
where $left | cdot right |$ is the absolute value operation and $j = sqrt{-1}$.






probability statistics absolute-value expected-value







share|cite|improve this question









$endgroup$




Let $varphi_i$ be a Gaussian random variable such that
$$varphi_i sim N(0,sigma^2), quad i = 1,2,ldots,n.$$
What's the expectation:
$$Eleft(left | sum_{i=1}^n e^{j varphi_i} right |right) $$
where $left | cdot right |$ is the absolute value operation and $j = sqrt{-1}$.






probability statistics absolute-value expected-value




probability statistics absolute-value expected-value






share|cite|improve this question













share|cite|improve this question











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asked Jan 1 at 15:32









Gertsen YuanGertsen Yuan

84




84












  • $begingroup$
    I don't know if there is a nice closed form for the absolute value but for the square of the absolute value the expectation is $frac{n (n-1)}{exp left(sigma ^2right)}+n$.
    $endgroup$
    – JimB
    Jan 1 at 19:55


















  • $begingroup$
    I don't know if there is a nice closed form for the absolute value but for the square of the absolute value the expectation is $frac{n (n-1)}{exp left(sigma ^2right)}+n$.
    $endgroup$
    – JimB
    Jan 1 at 19:55
















$begingroup$
I don't know if there is a nice closed form for the absolute value but for the square of the absolute value the expectation is $frac{n (n-1)}{exp left(sigma ^2right)}+n$.
$endgroup$
– JimB
Jan 1 at 19:55




$begingroup$
I don't know if there is a nice closed form for the absolute value but for the square of the absolute value the expectation is $frac{n (n-1)}{exp left(sigma ^2right)}+n$.
$endgroup$
– JimB
Jan 1 at 19:55










1 Answer
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$begingroup$

The expectation you are interested in is equivalent to



$$Eleft[sqrt{n+2sum_{j=1}^{n-1}sum_{k=j+1}^n (cos varphi_j cos varphi_k+sinvarphi_j sinvarphi_k)}right]$$



I don't believe there is a nice closed-form solution for that expectation. So I think you'll need to perform simulations or numerically integrate to get solutions.



If you were interested in



$$Eleft[n+2sum_{j=1}^{n-1}sum_{k=j+1}^n (cos varphi_j cos varphi_k+sinvarphi_j sinvarphi_k)right]$$



then there is a closed-form solution:



$$n+n(n-1)e^{-sigma^2}$$






share|cite|improve this answer









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    1 Answer
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    1 Answer
    1






    active

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    0












    $begingroup$

    The expectation you are interested in is equivalent to



    $$Eleft[sqrt{n+2sum_{j=1}^{n-1}sum_{k=j+1}^n (cos varphi_j cos varphi_k+sinvarphi_j sinvarphi_k)}right]$$



    I don't believe there is a nice closed-form solution for that expectation. So I think you'll need to perform simulations or numerically integrate to get solutions.



    If you were interested in



    $$Eleft[n+2sum_{j=1}^{n-1}sum_{k=j+1}^n (cos varphi_j cos varphi_k+sinvarphi_j sinvarphi_k)right]$$



    then there is a closed-form solution:



    $$n+n(n-1)e^{-sigma^2}$$






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      The expectation you are interested in is equivalent to



      $$Eleft[sqrt{n+2sum_{j=1}^{n-1}sum_{k=j+1}^n (cos varphi_j cos varphi_k+sinvarphi_j sinvarphi_k)}right]$$



      I don't believe there is a nice closed-form solution for that expectation. So I think you'll need to perform simulations or numerically integrate to get solutions.



      If you were interested in



      $$Eleft[n+2sum_{j=1}^{n-1}sum_{k=j+1}^n (cos varphi_j cos varphi_k+sinvarphi_j sinvarphi_k)right]$$



      then there is a closed-form solution:



      $$n+n(n-1)e^{-sigma^2}$$






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        The expectation you are interested in is equivalent to



        $$Eleft[sqrt{n+2sum_{j=1}^{n-1}sum_{k=j+1}^n (cos varphi_j cos varphi_k+sinvarphi_j sinvarphi_k)}right]$$



        I don't believe there is a nice closed-form solution for that expectation. So I think you'll need to perform simulations or numerically integrate to get solutions.



        If you were interested in



        $$Eleft[n+2sum_{j=1}^{n-1}sum_{k=j+1}^n (cos varphi_j cos varphi_k+sinvarphi_j sinvarphi_k)right]$$



        then there is a closed-form solution:



        $$n+n(n-1)e^{-sigma^2}$$






        share|cite|improve this answer









        $endgroup$



        The expectation you are interested in is equivalent to



        $$Eleft[sqrt{n+2sum_{j=1}^{n-1}sum_{k=j+1}^n (cos varphi_j cos varphi_k+sinvarphi_j sinvarphi_k)}right]$$



        I don't believe there is a nice closed-form solution for that expectation. So I think you'll need to perform simulations or numerically integrate to get solutions.



        If you were interested in



        $$Eleft[n+2sum_{j=1}^{n-1}sum_{k=j+1}^n (cos varphi_j cos varphi_k+sinvarphi_j sinvarphi_k)right]$$



        then there is a closed-form solution:



        $$n+n(n-1)e^{-sigma^2}$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 2 at 0:46









        JimBJimB

        55537




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