A highly simplified view of copy number detection with microarray techniques. For this exercise, we assume that a DNA of an individual has been measured by a DNA microarray, such that different DNA segments correspond to probes that react with light intensity. Higher intensity indicates the presence of a DNA segment, lower intensity the absence of a DNA segment. We assume that the probes of a segment with normal copy number"(state 2) emit an intensity with Gaussian distribution with a mean 2=7.0 and standard deviation 2=1.0 A segment with a copy number deletioncorresponds to a probe with intensity of 1=5.5,1=2.0 and a segment with copy number amplification "with intensity 3=8.0 and 3=1.5. (HINT: note that these distributions are necessary to calculate pE(xtut) from the Lecture Notes). We will consider five probes that are located on consecutive DNA segments, such that the copy number in one segment depends on the copy number of the precursor segment. This means we will have observation vectors such as x=(8.70,6.64,10.27,9.83,6.61), where each of the entries corresponds to the measurement of one probe and the probes are located in this order on the DNA. We will model the data arising from these DNA segments with a hidden Markov model with three hidden states (S={1,2,3}) corresponding to the copy number status "deletion", normal", and "gain", where each state emits signals according to the above-mentioned Gaussian (normal distributions). The transition probabilities are given by A=0.380.050.010.600.900.390.020.050.60, where Aij corresponds to the probability that the state i transits to state j, thus the row sums are 1 . The initial state probabilities are pinit=(0.02,0.95,0.03)