By Hans-Michael Kaltenbach
The textual content provides a concise advent into basic suggestions in records. bankruptcy 1: brief exposition of likelihood thought, utilizing common examples. bankruptcy 2: Estimation in idea and perform, utilizing biologically inspired examples. Maximum-likelihood estimation in lined, together with Fisher info and tool computations. equipment for calculating self assurance periods and strong possible choices to straightforward estimators are given. bankruptcy three: speculation checking out with emphasis on innovations, fairly type-I , type-II error, and studying try out effects. numerous examples are supplied. T-tests are used all through, vital different checks and robust/nonparametric possible choices. a number of trying out is mentioned in additional intensity, and mixture of self sustaining exams is defined. bankruptcy four: Linear regression, with computations completely in accordance with R. a number of team comparisons with ANOVA are lined including linear contrasts, back utilizing R for computations.
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Median absolute deviation. The MAD follows the same ideas as the variance, but measures the median of the absolute distance to the median: MAD = mediani (|xi − median j (x j )|). Inter-quartile range. 75-quantile, and is given by the rectangle in a boxplot (see Sect. 4) that contains the medium 50% of the data: IQR = q 3 − q 1 . 4 4 Comparison to the variance. Both the MAD and the IQR give different measures for the scale compared to the variance. Similar to the median, they both are more based on the ranks and not the absolute values of the particular data, and have high breakdown points.
For k = 2 and θ = 2, the distribution has expectation μ = kθ = 4 and variance σ 2 = kθ 2 = 8; its density is shown in Fig. 8 (solid line). For comparison, a normal distribution with the same expectation and variance is plotted by a dashed line. As we can see, the density functions look very different, although both have the same mean and variance. For additionally capturing their different shapes, higher moments are needed (see Sect. 5). Example 8 Let us consider the following model of a random DNA sequence as introduced earlier: we assume independence among the nucleotides and in each position, the probabilities of having a particular nucleotide are p A , pC , pG , pT , respectively.
Importantly, the density f X (x) is not a probability. For a distribution of a random variable X, we can compute the α-quantile as the value qα such that P(X ≤ qα ) = α. Important discrete distributions are the Binomial and geometric distribution, important continuous distributions are the normal distribution, the exponential, and various statistical distributions, including the t-, F-, and χ 2 -distributions, which are all related. Several interesting properties of a probability distribution are given by its moments, some of which are the expectation, describing the location, the variance, describing the scale and the skewness and kurtosis, describing the asymmetry and heaviness of the tails, respectively.