All questions carry 10 marks and are mandatory.

You are given 6 2D data points with binary class labels. Start with a linear classifier y = sign(x1 x2) where x1, x2 are the two dimensions. Run Adaboost algorithm for 2 iterations, and thus obtain 2 more linear classifiers along with their weights. Carry out ensemble classification of these points using the 3 linear classifiers (including initial one).

IDNo 1 2 3 4 5 6

X1 5.3 2.8 3.1 4.5 4.2 6.7

X2 3.8 5.4 5.8 5.3 11.6 6.2

Y 1 -1 -1 1 -1 -1

You are studying a process where the readings always lie in the range [0,1], under two possible conditions. You cannot observe the condition behind each reading, but have the prior knowledge that both conditions are equally likely. From each reading, you need to figure out the corresponding condition. Considering the reading as a random variable, you know that it follows the following probability distribution under the two conditions:

Under condition 1, the reading may be any value (uniformly) between 0 and 0.7 with probability 0.5, and any value (uniformly) between 0.7 and 1 with probability 0.5.

Under condition 2, the reading may be any value x between 0,1 with probability proportional to x

a)Calculate the probability density functions of the distributions under both conditions

b)Pose the above situation as a Bayes Classifier, using the PDFs as class-conditional. What are the predictions and confidence by the classifier for different readings?

c)The Bayes Error is the total probability of making a wrong prediction (irrespective of the reading value). Calculate the Bayes Error, i.e. probability of predicting the wrong condition, in case of your Bayes classifier.