Consider a two-category classification problem with one-dimensional Gaussian distributions p(x|i) N(i , 2 ), i = 1, 2 (i.e. they have same variance but different means).

i) Sketch the two densities p(x|1) and p(x|2) in one figure.

ii) Sketch the two posterior probabilities p(1|x) and p(2|x) in one figure, assuming same prior probabilities.

Also Consider a two-category classification problem based on samples with priors P(1) = P(2) = 0.5 and the distributions

p(x|1) = 2x , when 0 x 1 and 0, otherwise and p(x|2) = 2 2x, when 0 x 1 and 0, otherwise

i) Determine the Bayes decision rule.

ii) Calculate the Bayes classification error