Ex. 17.4
Ex. 17.4
Denote by \(f(X_1|X_2, X_3,...,X_p)\) the conditional density of \(X_1\) given \(X_2, X_3,...,X_p\). If
\[\begin{equation}
f(X_1|X_2,X_3,...,X_p) = f(X_1|X_3,...,X_p),\non
\end{equation}\]
show that \(X_1\bot X_2|X_3,...,X_p\).
Soln. 17.4
It's easy to see that
\[\begin{eqnarray}
f(X_1, X_2|X_3,...,X_p) &=& f(X_1|X_2,X_3,...,X_p) f(X_2|X_3,...,X_p)\non\\
&=&f(X_1|X_3,...,X_p)f(X_2|X_3,...,X_p).\non
\end{eqnarray}\]
Therefore we have \(X_1\bot X_2|X_3,...,X_p\).