Ex. 10.3

Show that the marginal average (10.47) recovers additive and multiplicative functions (10.50) and (10.51), while the conditional expectation (10.49) does not.

Soln. 10.3

The marginal average (10.47) is defined as

\[\begin{equation} f_S(X_S) = E_{X_C}f(X_S, X_C).\non \end{equation}\]

Note that it's different from the conditional expectation (10.49)

\[\begin{equation} \label{eq:10cond} \tilde f_X(X_S) = E[f(X_S, X_C)|X_S]. \end{equation}\]

Assuming the marginal density for \(X_C\) is \(\phi\). When \(f(X) = h_1(X_S) + h_2(X_C)\), we have

\[\begin{eqnarray} f_S(X_S) &=& \int f(X_S, c)\phi(c)dc\non\\ &=&\int [h_1(X_S) + h_2(c)]\phi(c)dc\non\\ &=&\int h_1(X_S)\phi(c)dc + \int h_2(c)\phi(c)dc\non\\ &=&h_1(X_S)\int\phi(c)dc + \int h_2(c)\phi(c)dc\non\\ &=&h_1(X_S) + \int h_2(c)\phi(c)dc\non \end{eqnarray}\]

where the last equation comes by noting \(\int \phi(c)dc = 1\). Similar arguments apply to \(f(X) = h_1(X_S)\cdot h_2(X_C)\).

However for the conditional expectation \(\eqref{eq:10cond}\), when \(f(X) = h_1(X_S) + h_2(X_C)\) we get

\[\begin{equation} \tilde f_S(X_S) = h_1(X_S) + E[h_2(X_C)|X_S].\non \end{equation}\]

When \(f(X) = h_1(X_S)\cdot h_2(X_C)\), we get

\[\begin{equation} \tilde f_S(X_S) = h_1(X_S)\cdot E[h_2(X_C)|X_S].\non \end{equation}\]