Problem 1. Let p(xw_{i}) ∼ N (µ_{i}, σ2) for a twocategory onedimensional problem with P (ω1) = P (ω2) = 2
Show that the minimum probability of error is given by where a = ^{}^{µ}1 −µ2  .
P_{e} = √2π
e−u /2du
P_{e} = √2πe−t /2 1dt ≤ √2πae−a /2
show that P_{e} goes to zero as ^{}^{µ}1 −µ2  goes to infinity.
Problem 2. Consider a twocategory classification problem in two dimensions with(xω1) ∼ N (0, I), p(xω2) ∼ N (.1Σ , I)and P (ω1) = P (ω2) = 1 ,
Problem 3. Suppose that we have three categories in two dimensions with the following underlying distributions:

p(x ω ) ( 1 , I) 1
0.3
of error.
Problem 4. Consider two normal distributions with arbitrary but equal covariances. Prove that the Fisher linear discriminant, for suitable threshold, can be derived from the negative of the loglikelihood ratio
Problem 5. It is easy to see that the nearestneighbor error rate P can equal the Bayes rate P ^{∗} if P ^{∗} = 0 (the best possibility) or if P ^{∗} = ^{c}^{−}1 (the worst possibility). One might ask whether or not there are problems for which P = P ^{∗} when P ^{∗} is between these extremes.
Problem 6. Prove that the computational complexity of the basic nearestneighbor editing algorithm for
n points in d dimension is O(d3n^{} d ∫ ln n).
NearestNeighbor Editing Algorithm
1: begin initialize j 0, data set, n #prototypes
2: construct the full Voronoi diagram of 3: do j j + 1, for each prototype x³ j 4: find the Voronoi neighbors of x³ j
5: if any neighbor is not from the same class as x³ j , then mark x³ j
6: until j = n
7: discard all points that are not marked
8: construct the Voronoi diagram of the remaining (marked) prototypes
9: end
Problem 7. Consider classifiers based on samples with priors P (ω1) = P (ω2) = 0.5 and the distributions
p(xω ) = .2x, 0 ≤ x ≤ 1
(2)0, elsewhere
p(xω ) = .2 − 2x, 0 ≤ x ≤ 1
elsewhere
Pn(e) = 3 + (n + 1)(n + 3) + 2(n + 2)(n + 3)
Confirm this formula makes sense in the n = 1 case.
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