Question

Expectation of a product by conditioning. Let X and Y be random variables, and let h be a function of X. Show that:

E [h(X)Y] = E [h(X)E(Y|X)]

(Hint: Look at E(h(X)Y|X = x).)
Remark: This identity, for indicator functions h(r), is used in more advanced treatments of probability to define conditional expectations given a continuous random variable X.

Answer 1

To show the equation E[h(X)Y] = E[h(X)E(Y|X)], we need to consider the conditional expectation E(h(X)Y|X = x) and apply the law of iterated expectations.

Step-by-step explanation:

To show the equation E[h(X)Y] = E[h(X)E(Y|X)], we need to consider the conditional expectation E(h(X)Y|X = x) and apply the law of iterated expectations.

1. 
   
2. Start by considering E(h(X)Y|X = x), which represents the conditional expectation of h(X)Y given X = x. This can be written as E(Y|X = x) multiplied by h(x) since h(X) does not depend on X = x.
3. 
   
4. Then, take the expectation on both sides of the equation above, resulting in E[E(h(X)Y|X = x)] = E[h(X)E(Y|X)].
5. 
   
6. By applying the law of iterated expectations, E[E(h(X)Y|X = x)] = E[h(X)Y], and the equation becomes E[h(X)Y] = E[h(X)E(Y|X)].