By Andrzej Polanski

ISBN-10: 3540241663

ISBN-13: 9783540241669

This textbook provides mathematical types in bioinformatics and describes organic difficulties that motivate the pc technology instruments used to regulate the large facts units concerned. the 1st a part of the booklet covers mathematical and computational equipment, with useful functions awarded within the moment half. The mathematical presentation avoids pointless formalism, whereas final transparent and certain. The booklet closes with an intensive bibliography, achieving from vintage examine effects to very fresh findings. This quantity is fitted to a senior undergraduate or graduate path on bioinformatics, with a robust concentrate on mathematical and desktop technological know-how background.

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59) is called the ﬁnite, discrete Jensen’s inequality. 59) remains valid. It is also possible to replace the discrete probability distribution containing atoms p1 , p2 , . . 6 The Expectation Maximization Method g[E(X)] ≤ E[g(X)]. 61) In the above X is a random variable with a probability density function f (x). 61), valid for every convex function g(x), is called the continuous Jensen’s inequality. 63) where again g(x) is a convex function and h(x) is any measurable function. 61) when we substitute Y = h(X).

39). When a > 1, b > 1 the graph is bell shaped; when a < 1, b < 1 the graph is U-shaped. 39) describes uniform distribution over the interval (0, 1). The moments of a random variable X described by the beta distribution are a ab E(X) = , Var(X) = . a+b (a + b)2 (a + b + 1) Its characteristic function is given by a sum of a hypergeometric series; we shall not give its exact form here. 4 Likelihood maximization It is a frequent situation that we try to determine from what distribution the data at our disposal were sampled.

If FX (x) is diﬀerentiable, then its derivative is called the probability density function (pdf) fX (x), where dFX (x) F (x < X ≤ x + Δx) = . 7) x −∞ fX (ξ)dξ = FX (x), and consequently, since FX (+∞) = 1, we obtain the normalization condition for the distribution of the continuous random variable X, +∞ −∞ fX (x)dx = lim FX (x) = 1. 1 Vector Random Variables It is often necessary to analyze distributions of two or more random variables jointly, which leads to vector random variables. For discrete random variables X assuming values x0 , x1 , x2 , .

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