Let X, Y, Z be independent jointly continuous random variables, and let g, h, r be some functions. For each one of the following formulas, state whether it is true for all choices of the functions g, h, and r or false (i.e., not true for all choices of these functions) Do not attempt formal derivations, use an intuitive argument

Question

1 E[g(X, Y) * h(Z)] = E[g(X, Y)] * E[h(Z)]

Select an option

2. E[g(X, Y) * h(Y, Z)] = E[g(X, Y)] * E[h(Y, Z)]

Select an option

3. E[g(X) * r(Y) * h(Z)] = E[g(X)] * E[r(Y)] * E[h(Z)]

Select an option

Answer 1

To determine whether each formula is true or false for all choices of the functions \( g, h, \) and \( r \), let's analyze each statement intuitively.

1. \( E[g(X, Y) * h(Z)] = E[g(X, Y)] * E[h(Z)] \)

   This statement is true. When \( Z \) is independent of \( X \) and \( Y \), the expectation of the product of \( g(X, Y) \) and \( h(Z) \) factors into the product of their individual expectations. The functions \( g \) and \( h \) can be any functions, and this independence property holds regardless of their specific forms.

2. \( E[g(X, Y) * h(Y, Z)] = E[g(X, Y)] * E[h(Y, Z)] \)

   This statement is false. Although \( X \), \( Y \), and \( Z \) are independent, \( h(Y, Z) \) depends on \( Y \). In this case, \( g(X, Y) \) and \( h(Y, Z) \) are not independent. The dependence of \( h \) on \( Y \) means that the expectation of the product does not generally split into a product of expectations.

3. \( E[g(X) * r(Y) * h(Z)] = E[g(X)] * E[r(Y)] * E[h(Z)] \)

   This statement is true. Since \( g(X) \), \( r(Y) \), and \( h(Z) \) depend on independent random variables \( X \), \( Y \), and \( Z \) respectively, the expectation of the product factors nicely into the product of the individual expectations.

In summary:

1. True
2. False
3. True