136 CH A P T E R 7: Inferring a Binomial Proportion via the Metropolis Algorithm In Equation 7.5, the summation is over values i sampled from p( ), and N is the number of sampled values. The approximation gets better as N gets larger. My use of limits in the summation notation in Equation 7.5 is unconventional, because the usual notation would indicate N instead of N p() . The uncon- i i ventional notation is extremely helpful, in my opinion, because it explicitly indicates the distribution of the discrete i values, and shows how the distribu- tion p( ) on the left side of the equation has an influence on the right side of the equation. Equation 7.5 is a special case of a general principle. For any function f ( ), the integral of that function, weighted by the probability distribution p( ), is approximated by the average of the function values at the sampled points. In math, d f ( ) p( ) 1 N N f ( i ) i p() (7.6) The approximation of the mean in Equation 7.5 merely has f ( ) = . Equa- tion 7.6 is a workhorse for the remainder of the chapter. You should think