Equivalent summations

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If the function $p(r)$ maps from the positive integers to the non-negative real numbers and has the property that $sum_r=1^infty p(r) = 1$, and $x_1, x_2, ... x_n$ is a sequence for which $X = sum_r=1^infty x_r p(r)$ is well-defined and the summation $sum_r=1^infty x_r^2p(r)$ is well defined, which of the following equals $sum_r=1^infty (x_r - X)^2p(r)$?



1)$[sum_r=1^infty (x_r^2 p(r)]-X^2$



2) $sum_r=1^infty (x_r^2 + X^2)p(r)$



3) $sum_r=1^infty (x_r^2 +2x_r X - X^2)p(r)$



4) $sum_r=1^infty (x_r - X)^2p(r)$ may not be well defined




summation




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    If the function $p(r)$ maps from the positive integers to the non-negative real numbers and has the property that $sum_r=1^infty p(r) = 1$, and $x_1, x_2, ... x_n$ is a sequence for which $X = sum_r=1^infty x_r p(r)$ is well-defined and the summation $sum_r=1^infty x_r^2p(r)$ is well defined, which of the following equals $sum_r=1^infty (x_r - X)^2p(r)$?



    1)$[sum_r=1^infty (x_r^2 p(r)]-X^2$



    2) $sum_r=1^infty (x_r^2 + X^2)p(r)$



    3) $sum_r=1^infty (x_r^2 +2x_r X - X^2)p(r)$



    4) $sum_r=1^infty (x_r - X)^2p(r)$ may not be well defined




    summation




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      If the function $p(r)$ maps from the positive integers to the non-negative real numbers and has the property that $sum_r=1^infty p(r) = 1$, and $x_1, x_2, ... x_n$ is a sequence for which $X = sum_r=1^infty x_r p(r)$ is well-defined and the summation $sum_r=1^infty x_r^2p(r)$ is well defined, which of the following equals $sum_r=1^infty (x_r - X)^2p(r)$?



      1)$[sum_r=1^infty (x_r^2 p(r)]-X^2$



      2) $sum_r=1^infty (x_r^2 + X^2)p(r)$



      3) $sum_r=1^infty (x_r^2 +2x_r X - X^2)p(r)$



      4) $sum_r=1^infty (x_r - X)^2p(r)$ may not be well defined




      summation




      share|cite|improve this question











      If the function $p(r)$ maps from the positive integers to the non-negative real numbers and has the property that $sum_r=1^infty p(r) = 1$, and $x_1, x_2, ... x_n$ is a sequence for which $X = sum_r=1^infty x_r p(r)$ is well-defined and the summation $sum_r=1^infty x_r^2p(r)$ is well defined, which of the following equals $sum_r=1^infty (x_r - X)^2p(r)$?



      1)$[sum_r=1^infty (x_r^2 p(r)]-X^2$



      2) $sum_r=1^infty (x_r^2 + X^2)p(r)$



      3) $sum_r=1^infty (x_r^2 +2x_r X - X^2)p(r)$



      4) $sum_r=1^infty (x_r - X)^2p(r)$ may not be well defined





      summation





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      asked Jul 26 at 4:27









      sanjayr

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          The answer is 2). Just expand $(x_r -X)^2$ as $x_r^2+X^2-2x_rX$ and you will get 2) easily.






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            The answer is 2). Just expand $(x_r -X)^2$ as $x_r^2+X^2-2x_rX$ and you will get 2) easily.






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              The answer is 2). Just expand $(x_r -X)^2$ as $x_r^2+X^2-2x_rX$ and you will get 2) easily.






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                up vote
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                The answer is 2). Just expand $(x_r -X)^2$ as $x_r^2+X^2-2x_rX$ and you will get 2) easily.






                share|cite|improve this answer













                The answer is 2). Just expand $(x_r -X)^2$ as $x_r^2+X^2-2x_rX$ and you will get 2) easily.







                share|cite|improve this answer













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                answered Jul 26 at 5:42









                Kavi Rama Murthy

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