By Andrzej Polanski

This textbook provides mathematical types in bioinformatics and describes organic difficulties that motivate the pc technology instruments used to regulate the large information units concerned. the 1st a part of the e-book covers mathematical and computational equipment, with useful functions awarded within the moment half. The mathematical presentation avoids pointless formalism, whereas final transparent and distinct. The publication closes with a radical bibliography, attaining from vintage examine effects to very fresh findings. This quantity is suited to a senior undergraduate or graduate path on bioinformatics, with a powerful specialise in mathematical and desktop technological know-how background.

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1 The Idea Suppose we wanted to verify whether a coin was symmetric and we tossed it 50 times. As a result we observed 50 heads. Although this result of the experiment is not inconceivable, we would definitely not believe in the symmetry of the coin. In other words, we would reject the hypothesis of a symmetry. But what if we saw 10 heads and 40 tails? Is there a premise for rejecting the hypothesis of symmetry of the coin? 46 2 Probability and Statistics Fig. 5. Illustration of the construction of the critical region for the binomial test.

In such a situation, the Wilcoxon sign-rank test can be applied. We need to compute the following Wilcoxon statistics. For each student, we compute the difference between the two sores, Di . , r(|DK |). We also keep a record of the original signs of the differences, and we denote by I + the list of indices i for which the signs were positive and we denote by I − the complementary list of negative signs. The Wilcoxon statistic T + is defined as r(|Di |). 91) i∈I + The null hypothesis is that the afternoon study program does not improve (change) the distribution of test scores.

6 The Expectation Maximization Method 37 Rao–Blackwell Theorem This theorem shows how one can improve estimators of parameters by applying sufficient statistics. Denote by pˆ any estimator of a parameter p, given observations X1 , X2 , . . , XN , and define a new estimator pˆnew as the conditional expectation p|t(X1 , X2 , . . , XN )] pˆnew = E[ˆ where t(X1 , X2 , . . , XN ) is a sufficient statistic for p. The Rao–Blackwell theorem states that E[(ˆ pnew − p)2 ] ≤ E[(ˆ p − p)2 ]. Consider the uniform distribution examined earlier in Sect.

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