I would like to know whether I applied the law of iterated expectation correctly.
- $E[X|A=a]=ca$ where $X_i$ are discrete random variable ($X_i>0$).
- $Y_i$ follows Binomial Distribution $Binomial(X,p)$
Then, is the following correct statement?
$E[Y|A=a]=E[E[Y|X]|A=a]=E[Xp|A=a]=cap$
I think: if $\{A=a\}\in \sigma(X)$ it is true.
if $\{A=a\}\in \sigma(X)$ so $E(Y1_{A=a}|X)=1_{A=a}E(Y|X) \hspace{.7cm} (1)$
by definition: $E(Z|B)=\frac{E(Z1_{B})}{P(B)}$
so
$$E(Y|A=a)=\frac{E(Y1_{A=a})}{P(A=a)}=\frac{E(E(Y1_{A=a})|X)}{P(A=a)}$$
$$\overset{(1)}{=}\frac{E(1_{A=a}E(Y|X))}{p(A=a)}=\frac{E(1_{A=a}Xp)}{p(A=a)}= p\frac{E(X1_{A=a})}{p(A=a)}=pE(X|A=a)=pca$$
so it depend of what is $\{A=a\}$?