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Let $C = A + B$ (statistical sum, so $\mathbb{E}[C] = \mathbb{E}[A] + \mathbb{E}[B]$), and let $p(A = a) = 1$. Are the following true?

  1. $\mathbb{E}[C^2] = a^2 + 2a\mathbb{E}[B] + \mathbb{E}[B^2]$
  2. $\mathbb{E}[C^3] = a^3 + 3a^2\mathbb{E}[B] + 3a\mathbb{E}[B^2] + \mathbb{E}[B^3]$
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This is only true if A and B are independent. In that case,

$\mathbb{E}[AB] = \mathbb{E}[A]\mathbb{E}[B]$ and your calculations are correct.

Without knowing the distribution for B, this is the strongest supposition needed for your calculations to be correct.

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4  
-1; Both identities are always true since A is non-random – Slaviks May 8 '13 at 14:03

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