Let's evaluate each statement one at a time.
Statement 1:
\[ E[g(Y)| X = x ]= \int g(y) f_{Y|X}(y|x) , dy \]
This statement is True.
Explanation: The formula states that the expected value of a function \(g(Y)\), given that \(X=x\), is equal to the integral of \(g(y)\) weighted by the conditional probability density function (PDF) of \(Y\) given \(X\). This is indeed a standard result in probability theory and holds for all choices of the function \(g\).
Statement 2:
\[ E[g(y)| X = x ]= \int g(y) f_{Y|X}(y|x) , dy \]
This statement is False.
Explanation: The expected value \(E[g(Y) | X = x]\) is a function of \(Y\) given the specific value of \(X=x\). Therefore, the left-hand side is not correctly represented by just \(g(y)\) but rather by the random variable \(g(Y)\), which must be evaluated conditionally based on the distribution of \(Y\) given \(X=x\). The correct form should actually reflect the expectation with respect to the conditional distribution of \(Y\) given \(X=x\). The expected value of \(g(Y)\) would depend on the distribution of \(Y\) and not just substitute \(y\) in \(g(y)\). Thus, it doesn't work for all choices of \(g\).
In summary:
- True
- False