Problem 3

a) The area under each triangle must equal 1. Both triangles have the same base, b = 2. Therefor k = 1 for both pdfs.
b) The decision point will be where they intersect at 2.5. Everything less than 2.5 will be classified as from ω_1and everything greater than 2.5 will be classified as ω_2.
c) Sketch a graph that indicates the Bayes error from the classifier in (b). Mark the important points.

RowBox[{RowBox[{tops, , =, , RowBox[{ListPlot, [, RowBox[{RowBox[{{, RowBox[{RowBox[{{, RowB ... wBox[{{2, 0}, ,, {3, 0}, ,, RowBox[{{, RowBox[{2.5, ,, 0.25}], }}]}], }}], ]}]}], }}], ]}]}], ;}]

RowBox[{RowBox[{Show, [, RowBox[{tops, ,, dist1, ,, dist2, ,, err, ,, errtop, ,, decision, ,, ... 2, ,, 2.5, ,, 3, ,, 4}], }}], ,, RowBox[{{, RowBox[{0, ,, 0.25, ,, 0.5}], }}]}], }}]}]}], ]}], ;}]

[Graphics:../HTMLFiles/index_16.gif]

Green Line → Top of triangles
Cyan Line → Top of Error triangle
Magenta Line → Decision Boundary
Red Triangle → p(x | ω_1) P(ω_1)
Blue Triangle → p(x | ω_2) P(ω_2)
Yellow Triangle → Bayes Error

d) Bayes error = 1/2 · 1 · 1/4 = 1/8 .

Created by Mathematica  (October 8, 2003)