Identify property of expectation

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I am unable to figure out which property of expectation is used in the following expression:



$E[(H^TH)^-1H^TWW^TH(H^TH)^-1]=E_WE_W[(H^TH)^-1H^TWW^TH(H^TH)^-1]$



$H,W$ are matrices of random variables. I found this in a book on statistical signal processing but cannot find anywhere how $E$ has been broken down into $E_WE_W$. Please help.






probability







share|cite|improve this question























  • Maybe it is just the law of total expectation? Sorry I am not sure about those subscript notation anyway.
    – BGM
    Aug 31 at 4:47










  • @BGM $E_H$ represents the expectation with respect to probability density function of H
    – mathamity
    Aug 31 at 5:20










  • Are you sure it's $E_H/W$ and not $E_W$? And are they really in this order? $E_WE_W$ would make more sense.
    – joriki
    Aug 31 at 6:36






  • 1




    Please correct the question. The question should stand for itself and not rely on the comments in order to be understandable. There's an edit link underneath the question.
    – joriki
    Aug 31 at 9:52







  • 1




    @joriki I have corrected the question.
    – mathamity
    Aug 31 at 12:25














up vote
0
down vote

favorite












I am unable to figure out which property of expectation is used in the following expression:



$E[(H^TH)^-1H^TWW^TH(H^TH)^-1]=E_WE_W[(H^TH)^-1H^TWW^TH(H^TH)^-1]$



$H,W$ are matrices of random variables. I found this in a book on statistical signal processing but cannot find anywhere how $E$ has been broken down into $E_WE_W$. Please help.






probability







share|cite|improve this question























  • Maybe it is just the law of total expectation? Sorry I am not sure about those subscript notation anyway.
    – BGM
    Aug 31 at 4:47










  • @BGM $E_H$ represents the expectation with respect to probability density function of H
    – mathamity
    Aug 31 at 5:20










  • Are you sure it's $E_H/W$ and not $E_W$? And are they really in this order? $E_WE_W$ would make more sense.
    – joriki
    Aug 31 at 6:36






  • 1




    Please correct the question. The question should stand for itself and not rely on the comments in order to be understandable. There's an edit link underneath the question.
    – joriki
    Aug 31 at 9:52







  • 1




    @joriki I have corrected the question.
    – mathamity
    Aug 31 at 12:25












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I am unable to figure out which property of expectation is used in the following expression:



$E[(H^TH)^-1H^TWW^TH(H^TH)^-1]=E_WE_W[(H^TH)^-1H^TWW^TH(H^TH)^-1]$



$H,W$ are matrices of random variables. I found this in a book on statistical signal processing but cannot find anywhere how $E$ has been broken down into $E_WE_W$. Please help.






probability







share|cite|improve this question















I am unable to figure out which property of expectation is used in the following expression:



$E[(H^TH)^-1H^TWW^TH(H^TH)^-1]=E_WE_W[(H^TH)^-1H^TWW^TH(H^TH)^-1]$



$H,W$ are matrices of random variables. I found this in a book on statistical signal processing but cannot find anywhere how $E$ has been broken down into $E_WE_W$. Please help.






probability




probability






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 31 at 10:08

























asked Aug 31 at 4:33









mathamity

1




1











  • Maybe it is just the law of total expectation? Sorry I am not sure about those subscript notation anyway.
    – BGM
    Aug 31 at 4:47










  • @BGM $E_H$ represents the expectation with respect to probability density function of H
    – mathamity
    Aug 31 at 5:20










  • Are you sure it's $E_H/W$ and not $E_W$? And are they really in this order? $E_WE_W$ would make more sense.
    – joriki
    Aug 31 at 6:36






  • 1




    Please correct the question. The question should stand for itself and not rely on the comments in order to be understandable. There's an edit link underneath the question.
    – joriki
    Aug 31 at 9:52







  • 1




    @joriki I have corrected the question.
    – mathamity
    Aug 31 at 12:25
















  • Maybe it is just the law of total expectation? Sorry I am not sure about those subscript notation anyway.
    – BGM
    Aug 31 at 4:47










  • @BGM $E_H$ represents the expectation with respect to probability density function of H
    – mathamity
    Aug 31 at 5:20










  • Are you sure it's $E_H/W$ and not $E_W$? And are they really in this order? $E_WE_W$ would make more sense.
    – joriki
    Aug 31 at 6:36






  • 1




    Please correct the question. The question should stand for itself and not rely on the comments in order to be understandable. There's an edit link underneath the question.
    – joriki
    Aug 31 at 9:52







  • 1




    @joriki I have corrected the question.
    – mathamity
    Aug 31 at 12:25















Maybe it is just the law of total expectation? Sorry I am not sure about those subscript notation anyway.
– BGM
Aug 31 at 4:47




Maybe it is just the law of total expectation? Sorry I am not sure about those subscript notation anyway.
– BGM
Aug 31 at 4:47












@BGM $E_H$ represents the expectation with respect to probability density function of H
– mathamity
Aug 31 at 5:20




@BGM $E_H$ represents the expectation with respect to probability density function of H
– mathamity
Aug 31 at 5:20












Are you sure it's $E_H/W$ and not $E_W$? And are they really in this order? $E_WE_W$ would make more sense.
– joriki
Aug 31 at 6:36




Are you sure it's $E_H/W$ and not $E_W$? And are they really in this order? $E_WE_W$ would make more sense.
– joriki
Aug 31 at 6:36




1




1




Please correct the question. The question should stand for itself and not rely on the comments in order to be understandable. There's an edit link underneath the question.
– joriki
Aug 31 at 9:52





Please correct the question. The question should stand for itself and not rely on the comments in order to be understandable. There's an edit link underneath the question.
– joriki
Aug 31 at 9:52





1




1




@joriki I have corrected the question.
– mathamity
Aug 31 at 12:25




@joriki I have corrected the question.
– mathamity
Aug 31 at 12:25















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