Law of iterated expectations applied to a ratio

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Consider the random variables $Y, Z_1, Zequiv(Z_1,dots,Z_n)$ with $Z_1,...,Z_n$ i.i.d.



Is it true that
$$mathbb Eleft(fracYZ_1right)=mathbb Eleft(fracmathbb E(Ymid Z)Z_1 right)$$?



Does the relation hold if we remove i.i.d.?




probability probability-theory expectation conditional-expectation




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    up vote
    -2
    down vote

    favorite












    Consider the random variables $Y, Z_1, Zequiv(Z_1,dots,Z_n)$ with $Z_1,...,Z_n$ i.i.d.



    Is it true that
    $$mathbb Eleft(fracYZ_1right)=mathbb Eleft(fracmathbb E(Ymid Z)Z_1 right)$$?



    Does the relation hold if we remove i.i.d.?




    probability probability-theory expectation conditional-expectation




    share|cite|improve this question























      up vote
      -2
      down vote

      favorite









      up vote
      -2
      down vote

      favorite











      Consider the random variables $Y, Z_1, Zequiv(Z_1,dots,Z_n)$ with $Z_1,...,Z_n$ i.i.d.



      Is it true that
      $$mathbb Eleft(fracYZ_1right)=mathbb Eleft(fracmathbb E(Ymid Z)Z_1 right)$$?



      Does the relation hold if we remove i.i.d.?




      probability probability-theory expectation conditional-expectation




      share|cite|improve this question













      Consider the random variables $Y, Z_1, Zequiv(Z_1,dots,Z_n)$ with $Z_1,...,Z_n$ i.i.d.



      Is it true that
      $$mathbb Eleft(fracYZ_1right)=mathbb Eleft(fracmathbb E(Ymid Z)Z_1 right)$$?



      Does the relation hold if we remove i.i.d.?





      probability probability-theory expectation conditional-expectation





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Jul 24 at 7:55
























      asked Jul 24 at 7:23









      STF

      15319




      15319




















          1 Answer
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          Yes. Since $frac 1 Z_1$ is measurable with respect to $sigma (Z_1,Z_2,..,Z_n)$ we can write the right side as $$EEleft(frac Y Z_1mid Zright)=Eleft(frac Y Z_1right)$$






          share|cite|improve this answer























          • What if we remove I.I.d.? Thank you
            – STF
            Jul 24 at 7:55






          • 1




            No need for any independence here. I just used basic properties of conditional expectation valid for any random variables a s long as ll the expectation s exist.
            – Kavi Rama Murthy
            Jul 24 at 7:59










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          1 Answer
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          active

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          1 Answer
          1






          active

          oldest

          votes









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          oldest

          votes






          active

          oldest

          votes








          up vote
          1
          down vote



          accepted










          Yes. Since $frac 1 Z_1$ is measurable with respect to $sigma (Z_1,Z_2,..,Z_n)$ we can write the right side as $$EEleft(frac Y Z_1mid Zright)=Eleft(frac Y Z_1right)$$






          share|cite|improve this answer























          • What if we remove I.I.d.? Thank you
            – STF
            Jul 24 at 7:55






          • 1




            No need for any independence here. I just used basic properties of conditional expectation valid for any random variables a s long as ll the expectation s exist.
            – Kavi Rama Murthy
            Jul 24 at 7:59














          up vote
          1
          down vote



          accepted










          Yes. Since $frac 1 Z_1$ is measurable with respect to $sigma (Z_1,Z_2,..,Z_n)$ we can write the right side as $$EEleft(frac Y Z_1mid Zright)=Eleft(frac Y Z_1right)$$






          share|cite|improve this answer























          • What if we remove I.I.d.? Thank you
            – STF
            Jul 24 at 7:55






          • 1




            No need for any independence here. I just used basic properties of conditional expectation valid for any random variables a s long as ll the expectation s exist.
            – Kavi Rama Murthy
            Jul 24 at 7:59












          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          Yes. Since $frac 1 Z_1$ is measurable with respect to $sigma (Z_1,Z_2,..,Z_n)$ we can write the right side as $$EEleft(frac Y Z_1mid Zright)=Eleft(frac Y Z_1right)$$






          share|cite|improve this answer















          Yes. Since $frac 1 Z_1$ is measurable with respect to $sigma (Z_1,Z_2,..,Z_n)$ we can write the right side as $$EEleft(frac Y Z_1mid Zright)=Eleft(frac Y Z_1right)$$







          share|cite|improve this answer















          share|cite|improve this answer



          share|cite|improve this answer








          edited Jul 24 at 8:05









          Did

          242k23208442




          242k23208442











          answered Jul 24 at 7:34









          Kavi Rama Murthy

          20.2k2829




          20.2k2829











          • What if we remove I.I.d.? Thank you
            – STF
            Jul 24 at 7:55






          • 1




            No need for any independence here. I just used basic properties of conditional expectation valid for any random variables a s long as ll the expectation s exist.
            – Kavi Rama Murthy
            Jul 24 at 7:59
















          • What if we remove I.I.d.? Thank you
            – STF
            Jul 24 at 7:55






          • 1




            No need for any independence here. I just used basic properties of conditional expectation valid for any random variables a s long as ll the expectation s exist.
            – Kavi Rama Murthy
            Jul 24 at 7:59















          What if we remove I.I.d.? Thank you
          – STF
          Jul 24 at 7:55




          What if we remove I.I.d.? Thank you
          – STF
          Jul 24 at 7:55




          1




          1




          No need for any independence here. I just used basic properties of conditional expectation valid for any random variables a s long as ll the expectation s exist.
          – Kavi Rama Murthy
          Jul 24 at 7:59




          No need for any independence here. I just used basic properties of conditional expectation valid for any random variables a s long as ll the expectation s exist.
          – Kavi Rama Murthy
          Jul 24 at 7:59












           

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