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Semester 1: Distribution Theory
Functions of random variables and their distributions using Jacobian transformation
Functions of Random Variables and Their Distributions Using Jacobian Transformation
Introduction to Random Variables
A random variable is a numerical outcome of a random phenomenon. It can be discrete or continuous. Discrete random variables take specific values with certain probabilities, while continuous random variables can take any value in a range.
Functions of Random Variables
Functions of random variables involve creating new random variables by transforming existing ones through functions. If X is a random variable, then Y = g(X) is a new random variable defined by some function g. The probability distribution of Y can be derived from that of X.
Jacobian Transformation
The Jacobian transformation is a method used to find the distribution of a function of multiple random variables. It involves computing the Jacobian determinant, which represents the change of variables in multiple dimensions. For two transformations U = g1(X, Y) and V = g2(X, Y), the joint distribution of (U, V) can be obtained from (X, Y) using the Jacobian.
Distributions and Their Transformations
Common distributions such as Normal, Exponential, and Uniform can be transformed using the Jacobian method. The transformation can help in deriving the probability density function (PDF) of the new random variables from the original ones.
Applications of the Jacobian Transformation
The Jacobian transformation is used in various applications, including change of variables in statistics, multivariate analysis, and simulations. It is particularly useful when dealing with correlated random variables or when integrating over complex domains.
Example of Jacobian Transformation
For instance, consider transforming from Cartesian coordinates (X, Y) to Polar coordinates (R, Θ). The relationship is defined as R = √(X^2 + Y^2) and Θ = tan^{-1}(Y/X). The Jacobian determinant is computed to find the joint distribution of R and Θ.
Conclusion
Understanding functions of random variables and the use of Jacobian transformations is crucial in distribution theory, particularly in the field of statistics, as it provides a systematic approach to analyze complex random processes.
Bivariate Normal Distribution, Compound and truncated distributions
Bivariate Normal Distribution, Compound and Truncated Distributions
Bivariate Normal Distribution
Bivariate normal distribution describes the joint distribution of two continuous random variables that are both normally distributed. It is characterized by two means, two variances, and the correlation coefficient which defines the relationship between the two variables. The probability density function has a bell-shaped contour and can be visualized as a three-dimensional surface.
Properties of Bivariate Normal Distribution
Key properties include symmetry about the means, the marginal distributions being univariate normal, and the correlation coefficient determining the shape of the distribution. If the correlation coefficient is zero, the two variables are independent.
Application of Bivariate Normal Distribution
This distribution is useful in various fields such as finance, biology, and social sciences to model relationships between two related continuous outcomes.
Compound Distributions
Compound distributions are formed by combining two or more random processes. They are often used to model scenarios where the number of occurrences of an event (e.g., claims in insurance) can vary and is itself random.
Characteristics of Compound Distributions
Key characteristics include the ability to capture a variety of behaviors by mixing different distributions. Examples include compound Poisson distributions.
Applications of Compound Distributions
Commonly used in insurance, telecommunications, and queuing theory where events are aggregated over a given time period.
Truncated Distributions
Truncated distributions occur when we restrict the sample space of a distribution, discarding values beyond certain limits. This is important in real-life scenarios where we are only interested in a certain range of data.
Characteristics of Truncated Distributions
A truncated distribution is modified so that it only includes values above or below a certain threshold, which alters the probabilities of outcomes.
Applications of Truncated Distributions
Useful in econometrics and biostatistics, where data collected may not cover the entire range of possible values.
Sampling distributions, non-central chi-square, t and F distributions
Distribution Theory
Sampling Distributions
Sampling distributions refer to the probability distribution of a statistic obtained through a large number of samples drawn from a specific population. The Central Limit Theorem plays a crucial role here, stating that, irrespective of the population distribution, the sampling distribution of the sample means approaches a normal distribution as the sample size becomes large.
Non-Central Chi-Square Distribution
The non-central chi-square distribution is a generalization of the chi-square distribution that arises in scenarios where the data has a non-zero mean. It is characterized by non-centrality parameters and is significant in various statistical applications including power analysis and hypothesis testing.
t Distribution
The t distribution, also known as Student's t distribution, is vital in hypothesis testing and is used when the sample size is small, and the population standard deviation is unknown. The t distribution is symmetric and has heavier tails than the normal distribution, providing a more accurate estimation for smaller sample sizes.
F Distribution
The F distribution is used primarily in the context of variance analysis and regression analysis. It is the ratio of two scaled chi-square distributions and is utilized to test the equality of variances across different populations. The F distribution is always positive and is crucial in ANOVA and regression modeling.
Order statistics and their distributions, extreme value and asymptotic distributions
Order Statistics and Their Distributions
Introduction to Order Statistics
Order statistics are statistics obtained from the ordered values of a sample. For a given sample of size n, the k-th order statistic is the k-th smallest value. They are useful in various statistical analyses, particularly in estimating percentiles and providing insights into the distribution's properties.
Distribution of Order Statistics
The distribution of order statistics can be derived from the original sample distribution. The k-th order statistic from a sample of independent and identically distributed random variables has a known probability distribution. Such distributions can be utilized to understand the behavior and properties of sample extremes.
Extreme Value Theory
Extreme Value Theory (EVT) focuses on the statistical behavior of extreme values. EVT provides distributions for the maximum and minimum values in a sample. The main distributions are the Gumbel, Fréchet, and Weibull distributions, which model the behavior of extremes under different conditions.
Asymptotic Distributions
Asymptotic distributions describe the behavior of order statistics and other estimators as the sample size tends to infinity. Widespread results, such as the Central Limit Theorem, apply to the distributions of normalized order statistics and other statistics in large samples, enabling effective statistical inference.
Applications of Order Statistics
Order statistics are employed in various fields such as reliability engineering, environmental studies, and finance. They help in modeling extreme outcomes, making predictions, and conducting risk assessments.
