Statistics in geophysics introduction and probability theory
Statistics in geophysics introduction and probability theory
... 1000 1500 Days since January 1st, 2002 Winter Term 2013/14 7/32 2000 Introduction Probability Theory Course outline Probability Theory Descriptive Statistics Inferential Statistics Linear Regression ... 27/32 Introduction Probability Theory Set theory The meaning of probability Some properties of the probability function Multiplicative law of probability and independence Rearranging the definition ... experiment is in the set A We define A⊂B⇔x ∈A⇒x ∈B , A = B ⇔ A ⊂ B and B ⊂ A Winter Term 2013/14 11/32 Introduction Probability Theory Set theory The meaning of probability Some properties of the probability...
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Statistics in geophysics probability theory II
... heads obtained in the three tosses ω X (ω) HHH HHT HTH THH TTH THT HTT TTT Table: Enumeration of the value of X for each point in the sample space x P(X = x) 1/8 3/8 3/8 1/8 Table: Induced probability ... distribution describes the number of events occurring in a certain time interval and so pertains to data on counts A random variable X is defined to have a Poisson distribution, denoted by X ... Common Probability Distributions Multiple Random Variables Random variables In many experiments it is easier to deal with a summary variable than with the original probability structure Example: In...
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Báo cáo toán học: "Some inequalities in functional analysis, combinatorics, and probability theory" pdf
... The main purpose of this paper is to show that many inequalities in functional analysis, probability theory and combinatorics are immediate corollaries of (2) ... Caen-Selberg inequality to the study of the finite field Kakeya and Nikodym problems in classical analysis the electronic journal of combinatorics 17 (2010), #R58 2 Inequalities in Functional Analysis ... Nikodym sets in finite fields Preprint [13] J E Pe˘ari´, On some classical inequalities in unitary spaces Mat Bilten 42 (1992) c v 63–72 [14] A R´nyi, Probability Theory North-Holland Series in Applied...
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Introduction to Probability Theory
... were removed in the Cantor set construction of Example 3.2 CHAPTER Introduction to Probability Theory 35 In addition to tossing a coin, another common random experiment is to pick a number, perhaps ... X and the probability measure IP we use in If we set the probability of H to be , then LS2 assigns mass to the number 16 If we set the probability of H to be , then LS2 assigns mass to the number ... first tossg fH on the first two tossesg fH on the first three tossesg ; and n=1 fH on the first n tossesg = fH on every tossg: According to Remark 1.1(d) (continuity from above), IP fH on every tossg...
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an introduction to probability theory - geiss
... (x)p(x)dx = Ef with the Riemann-integral on the left-hand side and the expectation of the random variable f with respect to the probability space (R, B(R), P) on the right-hand side Let us consider ... L and An ⊆ An+ 1 , n = 1, 2, imply ∞ n=1 An ∈ L Proposition 1.4.2 [ - -Theorem] If P is a π-system and L is a λsystem, then P ⊆ L implies σ(P) ⊆ L Definition 1.4.3 [equivalence relation] An ... from ω to f (ω)? This yields to the introduction of the random variables in Chapter Step 3: What are properties of f which might be important to know in practice? For example the mean-value and...
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communication and probability theory
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mathematics - introduction to probability theory
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Statistics in geophysics descriptive statistics II
... −1 ≤ rSP ≤ Interpretation: rSP > 0: Y tends to increase when X increases rSP < 0: Y tends to decrease when X increases rSP = 0: No tendency for Y to either increase or decrease when X increases ... modified version The following quantities (called fences) can be used for identifying extreme values in the tails of the distribution: lower inner fence: x0.25 − 1.5 × IQR; upper inner fence: x0.75 + ... of the Pearson correlation is the covariance between X and Y in the numerator The denominator is in effect just a scaling constant Winter Term 2013/14 23/29 Numerical Summary Measures Boxplots...
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Statistics in geophysics descriptive statistics
... statistical ideas is in making sense of a set of data The goal is to extract insights about the processes underlying the generation of the numbers Descriptive statistics is the discipline of quantitatively ... finite or countable set of different values continuous: uncountable set of different values quasi-continuous: data are continuous but measured in a discrete way Winter Term 2013/14 6/32 Setting ... Term 2013/14 18/32 Setting the scene Frequency distributions Histograms The range of the data is divided into class intervals or bins The number of values falling into each interval is counted The...
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Statistics in geophysics generalized linear regression
... Generalized linear models Binary regression Count data regression Components of the classical linear model Generalized linear models (GLMs) are an extension of classical linear models ... substituted into the respective model and the total number of parameters should be increased to p + Winter Term 2013/14 13/25 Generalized linear models Binary regression Count data regression Binary regression ... xik In this specification, the response variables are assumed to be (conditionally) independent Winter Term 2013/14 14/25 Generalized linear models Binary regression Count data regression Binary...
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Statistics in geophysics inferential statistics II
... distribution-free) tests Winter Term 2013/14 2/27 Introduction to hypothesis testing Some commonly encountered parametric tests Sampling distribution The sampling distribution for a statistic (including the test ... it justify questioning, the claim? This problem fits neatly into the parametric setting of the binomial distribution Winter Term 2013/14 11/27 Introduction to hypothesis testing Some commonly ... unfavourable to the null distribution Winter Term 2013/14 9/27 Introduction to hypothesis testing Some commonly encountered parametric tests Confidence intervals: inverting hypothesis tests There is a...
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Statistics in geophysics inferential statistics III
... Paired samples Two independent samples Background Tests not requiring assumptions involving specific parametric distributions for the data or for the sampling distribution of the test statistics are ... X1 , , Xn , Y1 , , Ym in ascending order Define Vi = if the i-th order statistic belongs to the X sample if the i-th order statistic belongs to the Y sample Winter Term 2013/14 9/10 Nonparametric ... distribution Winter Term 2013/14 5/10 Nonparametric tests for location One-sample case Paired samples Two independent samples Wilcoxon signed-rank test for paired data We assume that the sampling situation...
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Statistics in geophysics inferential statistics
... defined by summing the arguments and then dividing by n Winter Term 2013/14 3/26 Estimation by Maximum Likelihood Properties of Estimators Confidence Intervals Background In 1921, R A Fisher pointed ... parameters Winter Term 2013/14 12/26 Estimation by Maximum Likelihood Properties of Estimators Confidence Intervals Evaluating estimators We have outlined reasonable techniques for finding out estimators ... strict inequality holding somewhere Winter Term 2013/14 19/26 Estimation by Maximum Likelihood Properties of Estimators Confidence Intervals Interval estimation So far, we have dealt with the point...
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Statistics in geophysics linear regression II
... nonlinear relationships between continuous covariates and the response within the scope of linear models Two simple methods for dealing with nonlinearity: Variable transformation; Polynomial regression ... 13.8 25 W Turkey weights in pounds, ages in weeks, and origin (Georgia (G); Virgina (V); Wisconsin (W)), of 13 Thanksgiving turkeys Winter Term 2013/14 6/28 Multiple linear regression model Parameter ... against H1 : βj = Winter Term 2013/14 17/28 Multiple linear regression model Parameter estimation Hypothesis testing and confidence intervals Model choice and variable selection Testing linear...
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