Mathematical statistics is the application of probability theory and other mathematical concepts to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques that are commonly used in statistics include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory.
Introduction Statistical data collection is concerned with the planning of studies, especially with the design of randomized experiments and with the planning of surveys using random sampling. The initial analysis of the data often follows the study protocol specified prior to the study being conducted. The data from a study can also be analyzed to consider secondary hypotheses inspired by the initial results, or to suggest new studies. A secondary analysis of the data from a planned study uses tools from data analysis, and the process of doing this is mathematical statistics. Data analysis is divided into:
descriptive statistics – the part of statistics that describes data, i.e. summarises the data and their typical properties. inferential statistics – the part of statistics that draws conclusions from data (using some model for the data): For example, inferential statistics involves selecting a model for the data, checking whether the data fulfill the conditions of a particular model, and with quantifying the involved uncertainty (e.g. using confidence intervals). While the tools of data analysis work best on data from randomized studies, they are also applied to other kinds of data. For example, from natural experiments and observational studies, in which case the inference is dependent on the model chosen by the statistician, and so subjective.
Topics The following are some of the important topics in mathematical statistics:
Probability distributions
A probability distribution is a function that assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference. Examples are found in experiments whose sample space is non-numerical, where the distribution would be a categorical distribution; experiments whose sample space is encoded by discrete random variables, where the distribution can be specified by a probability mass function; and experiments with sample spaces encoded by continuous random variables, where the distribution can be specified by a probability density function. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures. A probability distribution can either be univariate or multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector—a set of two or more random variables—taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.
Special distributions Normal distribution, the most common continuous distribution Bernoulli distribution, for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no) Binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of independent occurrences Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs Geometric distribution, for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the negative binomial distribution, where the number of successes is one. Discrete uniform distribution, for a finite set of values (e.g. the outcome of a fair die) Continuous uniform distribution, for continuously distributed values Poisson distribution, for the number of occurrences of a Poisson-type event in a given period of time Exponential distribution, for the time before the next Poisson-type event occurs Gamma distribution, for the time before the next k Poisson-type events occur Chi-squared distribution, the distribution of a sum of squared standard normal variables; useful e.g. for inference regarding the sample variance of normally distributed samples (see chi-squared test) Student's t distribution, the distribution of the ratio of a standard normal variable and the square root of a scaled chi squared variable; useful for inference regarding the mean of normally distributed samples with unknown variance (see Student's t-test) Beta distribution, for a single probability (real number between 0 and 1); conjugate to the Bernoulli distribution and binomial distribution
Statistical inference
Statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation. Initial requirements of such a system of procedures for inference and induction are that the system should produce reasonable answers when applied to well-defined situations and that it should be general enough to be applied across a range of situations. Inferential statistics are used to test hypotheses and make estimations using sample data. Whereas descriptive statistics describe a sample, inferential statistics infer predictions about a larger population that the sample represents. The outcome of statistical inference may be an answer to the question "what should be done next?", where this might be a decision about making further experiments or surveys, or about drawing a conclusion before implementing some organizational or governmental policy. For the most part, statistical inference makes propositions about populations, using data drawn from the population of interest via some form of random sampling. More generally, data about a random process is obtained from its observed behavior during a finite period of time. Given a parameter or hypothesis about which one wishes to make inference, statistical inference most often uses:
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