I understand how you get from 1st to 2nd line but i don't get how the two terms are generated in the first line. I looked up Conditional Expectations but i still couldn't make sense of this. Thank you.
provenance...
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It seems that the order of the items being equated is not the best for understanding. It might be better to follow as $$ \mathrm{E}\{A_0\mid S(n)=1\}=\frac{\mathrm{P}\{A_0=1,S(n)=1\}}{\mathrm{P}\{S(n)=1\}} $$ by the definition of conditional probability. $$ \frac{\mathrm{P}\{A_0=1,S(n)=1\}}{\mathrm{P}\{S(n)=1\}}=\frac{\mathrm{E}\{A_0\cdot S(n)\}}{\mathrm{E}\{S(n)\}} $$ which follows if $A_0$ and $S(n)$ are independent and take values in $\{0,1\}$. $$ \frac{\mathrm{E}\{A_0\cdot S(n)\}}{\mathrm{E}\{S(n)\}}=\mathrm{E}\{A_0\}+\frac{\mathrm{E}\{A_0\cdot S(n)\}-\mathrm{E}\{A_0\}\cdot\mathrm{E}\{S(n)\}}{\mathrm{E}\{S(n)\}} $$ which is, of course, simply algebra. If your question was why the items were ordered as they are, I don't know. Perhaps the important equation that will be used later is $$ \mathrm{E}\{A_0\mid S(n)=1\}=\mathrm{E}\{A_0\}+\frac{\mathrm{E}\{A_0\cdot S(n)\}-\mathrm{E}\{A_0\}\cdot\mathrm{E}\{S(n)\}}{\mathrm{E}\{S(n)\}} $$ and so they wanted that to appear together. |
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If $\mathbb P(Y\in\{0,1\})=1$ and $X$ is integrable, then $\mathbb E(X\mid Y=1)=\mathbb E(XY)/\mathbb P(Y=1)$ and $\mathbb P(Y=1)=\mathbb E(Y)$ hence, for every real number $a$, $$ \mathbb E(X\mid Y=1)=\frac{\mathbb E(X;Y=1)}{\mathbb P(Y=1)}=\frac{\mathbb E(XY)}{\mathbb E(Y)}=a+\frac{\mathbb E(XY)-a\cdot\mathbb E(Y)}{\mathbb E(Y)}. $$ Apply this to $X=A_0$, $Y=S(n)$, and $a=E(A_0)$. |
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