Conditional expectation of a random variable conditioned to another conditionally independent variable
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I have a two variables $X$ and $Y$ conditionally independent given a thirs variable $Z$ . Now, assume that $Z$ can take values in ${1,...,k}$ . I will have: $$ E[XY] = sum_{i=1}^k P[Z = i]E[X|Z = i]E[Y|Z = i] $$ Now, can I say what follows? $$ E[X | Y] = E[X] $$ I have a proof for this, but I am not sure: $$ E[X | Y] = sum_{i=1}^k P[Z = i]E[X|Z = i, Y] = sum_{i=1}^k P[Z = i]E[X|Z = i] = E[X] $$ Is this correct? If not, where am I wrong?
probability conditional-expectation
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asked Dec 1 '18 at 15:56
Ulderique Demoitre Ulderique Demoitre