Question: A researcher wants to determine a procedure for discriminating between two multivariate populations. The researcher has enough data available to estimate the density functions \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\) associated with populations \(\pi_{1}\) and \(\pi_{2}\), respectively. Let \(c(2 \mid 1)=50\) (this is the cost of assigning items as \(\pi_{2}\), given that \(\pi_{1}\) is true) and \(c(1 \mid 2)=100\)

In addition, it is known that about \(20 \%\) of all possible items (for which the measurements \(\mathrm{x}\) can be recorded) belong to \(\pi_{2}\)

1. Give the minimum ECM rule (in general form) for assigning a new item to one of the two populations.
2. Measurements recorded on a new item yield the density values \(f_{1}(\mathbf{x})=.3\) and \(f_{2}(x)=.5 .\) Given the preceding information, assign this item to population \(\pi_{1}\) or population \(\pi_{2}\)

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