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Mathematical statistics 2 pdf. Mathematical statistics notes pdf. Mathematical statistics up. Mathematical statistics with applications. Mathematical statistics and data analysis. Mathematical statistics questions and answers pdf. Mathematical statistics syllabus. Mathematical statistics exercises and solutions.44k Accesses 421 Citations 18 Altmetric Page 2 Mathematical statistics relies on probability theory, which in turn is based on measure theory. The present chapter provides some principal concepts and notational conventions of probability theory, and some important results that are useful tools in statistics. A more complete account of probabilitytheory can be found in a standard textbook, for example, Billingsley (1986), Chung (1974), or Loève (1977). The reader is assumed to be familiar with set operations and set functions (mappings) in advanced calculus.KeywordsProbability MeasureIndependent Random VariableConditional ExpectationAsymptotic TheoryDominate ConvergenceTheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, access via your institution. Unable to display preview. Download preview PDF. We are delighted to report that the ICSDS (InternationalConference on Statistics and Data Science) on Dec. 13-16, 2022 at Florence, Italy has received tremendous responses, including many outstanding invited speakers coming from different countries and continents, covering a wide range of subjects in statistics and data science, in theory, methodology and applications. In particular, we are pleased toannounce the four confirmed plenary speakers: Emmanuel Candes, Guido Imbens, Susan Murphy and Sylvia Richardson, for This graduate textbook covers topics in statistical theory essential for graduate students preparing for work on a Ph.D. degree in statistics. This new edition has been revised and updated and in this fourth printing, errorshave been ironed out. The first chapter provides a quick overview of concepts and results in measure-theoretic probability theory that are useful in statistics. The second chapter introduces some fundamental concepts in statistical decision theory and inference. Subsequent chapters contain detailed studies on some important topics: unbiasedestimation, parametric estimation, nonparametric estimation, hypothesis testing, and confidence sets. A large number of exercises in each chapter provide not only practice problems for students, but also many additional results. Statistical Models: Elements of Decision Theory. Bayesian Models. Prediction. Sufficiency. Exponential Models. Methods ofEstimation: Minimum contrast estimates. Estimating equations. Weighted least squares. Empirical plug-in estimates. Maximum likelihood. Criteria: Minimax. Bayes. Unbiased. Information Inequality. Robustness. Testing and confidence regions: The Neyman-Pearson Lemma. Uniformly most Powerful Tests. Monotone Likelihood Ratio Models. TheDuality between Tests and Confidence Regions. Bayesian Formulations. Likelihood Ratio Procedures. Prediction Intervals. Asymptotic approximations. Consistency. The Delta Method. Asymptotic Normality of Estimates. Asymptotic efficiency of the maximum likelihood estimate. Statistics is a branch of mathematics that deals with the study ofcollecting, analyzing, interpreting, presenting, and organizing data in a particular manner. Statistics is defined as the process of collection of data, classifying data, representing the data for easy interpretation, and further analysis of data. Statistics also is referred to as arriving at conclusions from the sample data that is collected using surveys orexperiments. Different sectors such as psychology, sociology, geology, probability, and so on also use statistics to function. Mathematical Statistics Statistics is used mainly to gain an understanding of the data and focus on various applications. Statistics is the process of collecting data, evaluating data, and summarizing it into a mathematical form.Initially, statistics were related to the science of the state where it was used in the collection and analysis of facts and data about a country such as its economy, population, etc. Mathematical statistics applies mathematical techniques like linear algebra, differential equations, mathematical analysis, and theories of probability. There are two methodsof analyzing data in mathematical statistics that are used on a large scale: Descriptive Statistics Inferential Statistics Descriptive Statistics The descriptive method of statistics is used to describe the data collected and summarize the data and its properties using the measures of central tendencies and the measures of dispersion. Inferential StatisticsThis method of statistics is used to draw conclusions from the data. Inferential statistics requires statistical tests performed on samples, and it draws conclusions by identifying the differences between the 2 groups. Tests calculate the p-value that is compared with the probability of chance(α) 0.05. If the p-value is less than α, then it is concludedthat the p-value is statistically significant. Data Representation in Statistics The collection of observations and facts is known a data. These observations and facts can be in the form of numbers, measurements, or statements. There are two different kinds of data i.e. Qualitative data and quantitative data. Qualitative data is when the data is descriptiveor categorical and quantitative data is when the data is numerical information. Once we know the data collection methods, we aim at representing the collected data in different forms of graphs such as a bar graph, line graph, pie chart, stem and leaf plots, scatter plot, and so on. Before the analysis of data, the outliers are removed that are due to theinvariability in the measurements of data. Let us look at different kinds of data representation in statistics. Data Representation Description Bar Graph A group of data represented with rectangular bars with lengths proportional to the values is a bar graph. The bars can either be vertically or horizontally plotted. Pie Chart The pie chart is a type ofgraph in which a circle is divided into Sectors where each sector represents a proportion of the whole. Line graph The line graph represents the data in a form of series that is connected with a straight line. These series are called markers. Pictograph Data shown in the form of pictures is a pictograph. Pictorial symbols for words, objects, or phrasescan be represented with different numbers. Histogram The histogram is a type of graph where the diagram consists of rectangles, the area is proportional to the frequency of a variable and the width is equal to the class interval. Here is an example of a histogram. Frequency Distribution The frequency distribution table in statistics showcases the datain ascending order along with their corresponding frequencies. The frequency of the data is often represented by f. Different Models of Statistics Statistics being a broad term used in various forms, different models of statistics are used in different forms. Listed below are a few models: Skewness - In statistics, the word skewness refers to a measureof the asymmetry in a probability distribution where it measures the deviation of the normal distribution curve for data. The value of skewed distribution could be positive or negative or zero. The curve is said to be skewed when it shifts from left to right. If the curve mores towards the right it is called a positive skewed and if the curve moves towardsthe left, it is called left-skewed. ANOVA Statistics - The word ANOVA means Analysis of Variance. The measure used in calculating the mean difference for the given set of data is called the ANOVA statistics. This model of statistics is used to compare the performance of stocks over a period of time. Degrees of freedom - This model of statistics is usedwhen the values are changed. Data that can be moved while estimating a parameter is the degree of freedom. Regression Analysis - In this model, the statistical process determines the relationship between the variables. The process signifies how a dependent variable changes when an independent variable changed. Measures of Central Tendency inStatistics The measure of central tendency and the measure of dispersion are considered as the basis of descriptive statistics. The representative value for the given data is the measure of central tendency that gives us an idea of where data points are centered. This is done to find how the data are scattered around this centered measure. We usemean, median, and mode to find the central measures of tendency. In our day-to-day life, we find the average height of the students, the average income, the average score in exams, or of the player. The different measures of central tendency for the data are: Arithmetic Mean Median Mode Geometric Mean Harmonic Mean Mean, Median and Modein Statistics Mean is considered the arithmetic average of a data set that is found by adding the numbers in a set and dividing by the number of observations in the data set. The middle number in the data set while listed in either ascending or descending order is the median. Lastly, the number that occurs the most in a data set and ranges betweenthe highest and lowest value is the mode. For n number of observations, we have Mean \(\bar{x} \dfrac{\sum x}{n}\) Median \(\dfrac{n 1}{2}\)th term if n is odd. Median \(\dfrac{\dfrac{n}{2} {th} \text { term } (\dfrac{n}{2} 1) {th} \text {term}}{2}\) Mode The value which occurs most frequently Measures of Dispersion inStatistics The measures of central tendency do not suffice to describe the complete information about a given data. Thus we need to describe the variability by a value called the measure of dispersion. The different measures of dispersion are: The range in statistics is calculated as the difference between the maximum value and the minimum value ofthe data points. The quartile deviation that measures the absolute measure of dispersion. The data points are divided into 3 quarters. Find the median of the data points. The median of the data points to the left of this median is said to be the upper quartile and the median of the data points to the right of this median is said to be the lower quartile.Upper quartile - lower quartile is the interquartile range. Half of this is the quartile deviation. The mean deviation is the statistical measure to determine the average of the absolute difference between the items in a distribution and the mean or median of that series. The standard deviation is the measure of the amount of variation of a set of values.Mean Deviation For ungrouped data In statistics, the frequency distributions of data can be discrete data or continuous. For n number of individual observations \(x 1, x 2, x 3, x r, . x n\), the mean deviation about mean and median are calculated as follows: Mean Deviation for ungrouped data sum of deviation/number of observations \(\dfrac{\sum {i 1} {N} (x i -\bar{x})}{n}\) Mean Deviation for Discrete Grouped data The measurements of the data units are clearly shown in such a frequency distribution. Let there be n distinct data points \(x 1, x 2, x 3, x r, . x n\), occurring with frequencies \(f 1, f 2, f 3. f n\). a) Mean deviation about mean We find the mean \(\bar {x}\)using \(\dfrac{\sum {i 1} {N}(X {i}-f i)}{\sum {i 1} {N}f i}\). This is the ratio of the sum of the products of \(x i\) observations and their respective frequencies \(f i\) to the sum of the frequencies. Mean Deviation \(\dfrac{1}{N}\sum {i 1} {N}(x {i}f i)\) Afterwhich find the deviations of observations \(x i \) from the mean \(\bar {x}\) and gettheir absolute values. i.e. \(x i - \bar {x}\) for all i 1, 2, 3, .n Mean Deviation \(\bar {x} \dfrac{\sum {i 1} {N}f i\mid x i - \bar{x}\mid}{\sum {i 1} {N}f i}\) b) Mean deviation about median Find the median by arranging the observations in ascending order. Obtain the cumulative frequencies. Then identify the observation whosecumulative frequency is N/2, where N sum of frequencies. Thus we have arrived at the required median. To get the absolute values of the deviations from median, we calculate MD( median) \(\dfrac{1}{N}\sum {i 1} {N}f i \mid x i - M\mid\) Mean Deviation for Continuous Grouped data Here the data points take any value within a range andthey are continuous. They can be measured and represented by using intervals on the real number line. The frequency in which data are arranged in classes is not countable. a) Mean deviation about mean The mean of the continuous frequency distribution is centered at its mid-point in each class. Then the same procedure is followed as in the case ofdiscrete frequency distribution. b) Mean deviation about median Median \(l \dfrac{\dfrac{N}{2}-C}{f}\times h\), where the median class is the class interval whose cf is N/2, N the sum of frequencies, l, f, h, and C are, the lower limit, the frequency, the width of the median class and C the cumulative frequency of the class just preceding themedian class. After finding the median, \(x i\) - M is obtained. Standard Deviation and Variance We have the other prominent methods in statistics to find the proper measure of dispersion, known as the variance and the standard deviation. While finding the mean deviation about the mean and the median, there arises a difficulty in taking squares ofall the deviations. If \(\sum {i 1} {N}(x i -\bar x) 2\) becomes zero, while calculating the sum for the mean, then it means there is no dispersion at all. If the sum is small, the observations are closer to the mean indicating a lower degree of dispersion. If the sum is large, there is a higher degree of dispersion of the observations from the mean \(\barx\). Thus this sum is a reasonable indicator of the degree of dispersion. This becomes the proper measure of dispersion, denoted as σ2, and it is termed as the variance. Thus variance is given as σ2 \(\dfrac{\sum {i 1} {N}(x i -\bar x) 2}{N}\). The positive square root of the variance is called the standard deviation. σ \(\sqrt{\dfrac{\sum {i 1} {N}(x i -\bar x) 2}{N}}\). Coefficient of Variation We compare the coefficient of variations of two or more frequency distributions. This coefficient of variation in statistics is the ratio of the standard deviation to the mean, expressed in percentage. CV σ/ \(\bar {x}\) 100. The distribution that has a greater coefficient ofvariation has more variability around the central value than the distribution having a smaller value of the coefficient of variation. Important Notes The discipline of data collection and organization is called statistics. We interpret results based on the analysis done using the measures of central tendencies and the measures of dispersion. The frequencydistribution of data is represented using bar graphs, histograms, pie charts, stem and leaf plots, line graphs, or ogives. The data collected can be either quantitative (numerical: discrete and continuous) or qualitative(categorical). Also Check: Probability and Statistics Data Handling Example 1:Compute the mean deviation about mean from thefollowing data. Size(x) 2 4 6 8 10 Frequency f 2 4 5 3 1 Solution: In statistics, we know that the mean deviation about mean is calculated using the formla: Mean (Σ f x)/N [(2 2 4 4 6 5 8 3 10 1)/15 (4 16 30 24 10)/15 84/15 5.6 Answer: The mean deviation about mean 5.6 Example 2: The mean of 5 observations is 4.4and their variance is 8.24. If 3 of the observations are 1, 2, and 6, find the other two observations. Solution: Let the other two observations be a and b. Then we have (1 2 6 a b) /5 4.4 (1 2 6 a b) 22 a b 22 -9 13 a b 13 ----------- (1) Given : variance 8.24 We know that in statistics, variance is calculated as: σ2 \(\dfrac{\sum {i 1} {N}(x i -\bar x) 2}{N}\) 8.24 8.24 (1/5) [3.42 2.42 1.62 (a-4.4)2 (b-4.4)2 ] 41.20 11.56 5.46 2.56 a2 b2 2 4.4(a b) 2 4.42 41.20 19.58 38.72 114.4 a2 b2 a2 b2 97----------- (2) Form (1), we have (a b) 2 169----------- (3) Solving (2) and (3), we have 2 ab 72----------- (4) (4) - (2) givesa2 b2 -2 ab 169 - 72 25 Thus a - b 5 Substituting the values of 5 and - 5 in (1), we get a 9, b 4 (or) a 4, b 9 Answer: The other two observations are 4 and 9. Example 3: Find the standard deviation of 8,10,12,14,16. Solution: Given N 5 Σx 8 10 12 14 16 60 \(\bar {x}\) 60/5 12 In statistics, Standard deviation VarianceVariance σ 2 \(\dfrac{\sum {i 1} {N}(x i -\bar x) 2}{N}\). 1/5[(8-12)2 (10-12)2 (12-12)2 (14-12)2 (16-12)2] 1/5[16 4 0 4 16] 40/5 8 Standard deviation σ 8 2.83 Answer: The standard deviation of the given data 2.83 View More go to slidego to slidego to slide Great learning in high school using simple cuesIndulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes. Book a Free Trial Class FAQs on Statistics Statistics is a branch of mathematics that deals with the study of collecting, analyzing, interpreting, presenting, and organizing data in a particular manner. It is referred to asarriving at conclusions of data with the use of data. What are the Two Types of Statistics? The two different types of statistics are: Descriptive Statistics: It is used to summarize the data and its properties using mean and standard deviation. Inferential Statistics: It is used to get a conclusion from the data collected. What is Descriptive Statistics?Descriptive statistics describe the data features and provide summaries about the entire or sample population. We calculate the measures of central tendencies and measures of dispersion to summarize the data, in this type of statistics. What is Inferential Statistics? Inferential statistics predict and make inferences from the data is called inferentialstatistics. Many statistical tests are performed to arrive at conclusions. This inferential statistics has connections with probability and probability distribution. How is Statistics Used in Mathematics? Statistics is a part of applied mathematics that uses probability theory to simplify the sample data we collect. The concept of probability comes understatistics where we can determine if the data is true or false but mostly, the data is true. What is the Purpose of Statistics? Statistics helps in better understanding and accurate description. It also helps in proper planning in the statistical study. Finally, statistics uses tables, diagrams, and graphs as representing the information in a certain manner.What is the Importance of Statistics in Real Life? Statistics helps to utilize strategies to gather the information, examine them, and successfully present the outcomes. Measurement is a significant cycle behind how we make disclosures in science, settle on choices dependent on information, and make forecasts. What Are Examples of Statistics? Weconsider a class of students as a sample of the population of all the students in the school. We can calculate their average score in tests, their average height, weight etc based on the data collected. The required parameters are determined using the statistical measures are analyzed and interpreted further, as desired. For example, the scores of thestudents in the previous semester and this semester can be compared.

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Initially, statistics were related to the science of the state where it was used in the collection and analysis of facts and data about a country such as its economy, population, etc. Mathematical statistics applies mathematical techniques like linear algebra, differential equations, mathematical analysis, and theories of probability.