Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Continuous Distributions: Testing and Estimation

In course

MATH-236: Probability and statistics II

Linear statistical methods, analysis of experiments, logistic regression. ...

Description

This lecture covers the testing and estimation of continuous distributions, focusing on methods for partitioning intervals and calculating expected numbers. It includes examples of testing uniform and normal distributions, as well as parameter estimation using the chi-square test. The instructor explains how to modify test statistics for composite hypotheses and provides practical examples of hypothesis testing and parameter estimation.

Instructor

Darlene Goldstein

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics

Related concepts (32)

Normal distribution

In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter is its standard deviation. The variance of the distribution is . A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.

Continuous uniform distribution

In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, and which are the minimum and maximum values. The interval can either be closed (i.e. ) or open (i.e. ). Therefore, the distribution is often abbreviated where stands for uniform distribution.

Probability distribution

In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.

Normal-gamma distribution

In probability theory and statistics, the normal-gamma distribution (or Gaussian-gamma distribution) is a bivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and precision. For a pair of random variables, (X,T), suppose that the conditional distribution of X given T is given by meaning that the conditional distribution is a normal distribution with mean and precision — equivalently, with variance Suppose also that the marginal distribution of T is given by where this means that T has a gamma distribution.

Chi-squared test

A chi-squared test (also chi-square or χ2 test) is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are large. In simpler terms, this test is primarily used to examine whether two categorical variables (two dimensions of the contingency table) are independent in influencing the test statistic (values within the table). The test is valid when the test statistic is chi-squared distributed under the null hypothesis, specifically Pearson's chi-squared test and variants thereof.

Related lectures (33)

Probabilities and Statistics

Covers fundamental concepts in probabilities and statistics, including linear regression, exploratory statistics, and the analysis of probabilities.

Testing: t-tests

Covers t-tests, p-values calculation, and comparison of coefficients.

Estimation and Confidence Intervals

Explores bias, variance, and confidence intervals in parameter estimation using examples and distributions.

Linear Regression: Estimation and Testing

Explores linear regression estimation, hypothesis testing, and practical applications in statistics.

Continuous Random Variables