Conditional expectation

Description:

• Conditional joint distribution and Expectation
• Discrete case:
  • The conditional expectation of $X$ given $Y=y$, $E[X|Y=y]=\sum _{x}xp(X=x|Y=y)$
  • $E[g(X)|Y=y]=\sum _{x}g(x)p(X=x|Y=y)$
• Continuous case:
  • $E[X|Y=y]={\int }_{-\infty }^{\infty }x.f_{X|Y}(x|y)dx$
    • where $f_{X|Y}(x|y)=\frac{f(x,y)}{f_Y(y)}$ as in Conditional joint distribution
  • $E[g(X)|Y=y]={\int }_{-\infty }^{\infty }g(x).f_{X|Y}(x|y)dx$
• $E[\sum _{i=1}^n[x_i|Y=y]]=\sum _{i=1}^{n}E[X_i|Y=y]$
• Think of $E[X|Y]$ is itself a random variable. $E[X]=E[E[X|Y]]$ meaning:
  • For discrete: $E[X]=\sum _{y}E[X|Y=y].P(Y=y)$
    • To calculate $E[X]$, we may take a weighted average of the conditional expected value of $X$ given that $Y=y$, each of the terms $E[X|Y=y]$ being weighted by the probability of the event on which it is conditioned.
  • For continuous: $E[X]={\int }_{-\infty }^{\infty }E[X|Y=y].f_Y(y)dy$

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