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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?

$E[Y|A=a]=E[E[Y|X]|A=a]=E[Xp|A=a]=cap$

user3509199
  • 175

1 Answers1

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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 $\{A=a\}$?

Masoud
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