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Semester 2: Distribution Theory
Binomial distribution: moments, recurrence relation, moment generating function, fitting of binomial distribution
Binomial distribution
Definition and Properties
Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials with a constant probability of success. Key properties include: - The number of trials is denoted as n. - The probability of success on each trial is denoted as p. - The probability of failure is q = 1 - p. - The mean is np and the variance is npq.
Moments
Moments are statistical measures that provide information about the shape of the distribution. For the binomial distribution: - The first moment, or mean, is np. - The second moment is np(1 + p(n - 1)). - The variance is npq, which measures the spread of the distribution.
Recurrence Relation
The recurrence relation for the binomial coefficients can be expressed as: - C(n, k) = C(n-1, k-1) + C(n-1, k), where C(n, k) is the binomial coefficient representing the number of ways to choose k successes from n trials. This relation is useful for calculating probabilities in the binomial distribution.
Moment Generating Function
The moment generating function (MGF) of a binomial distribution is given by: - M(t) = (q + pe^t)^n, where q = 1 - p. The MGF is used to derive moments and other properties of the distribution.
Fitting of Binomial Distribution
Fitting a binomial distribution involves estimating parameters p and n from given data. Techniques include: - Method of moments where parameters are estimated by matching sample moments with theoretical moments. - Maximum likelihood estimation (MLE) is another approach that maximizes the likelihood function to obtain estimates of p and n.
Poisson distribution: moments, moment generating function, fitting of Poisson distribution
Poisson distribution
Introduction
The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, provided that these events occur with a known constant rate and are independent of the time since the last event.
Moments
The moments of a Poisson distribution can be derived from its probability mass function. The k-th moment about the origin is given by E[X^k] = λ^k, where λ is the rate parameter. The first moment (mean) is λ, and the second central moment (variance) is also λ.
Moment Generating Function (MGF)
The moment generating function of a Poisson distribution with parameter λ is given by M(t) = exp(λ(e^t - 1)). This function is useful for determining the moments of the distribution and can be used to prove properties of the Poisson distribution.
Fitting of Poisson Distribution
Fitting a Poisson distribution involves estimating the parameter λ from the data. This can be done using methods such as the method of moments, maximum likelihood estimation, or by using Poisson regression in case of multiple predictors. Goodness-of-fit tests can also be performed to validate the fitting.
Negative binomial distribution: moment generating function, cumulants
Negative binomial distribution
Definition
The negative binomial distribution is a discrete probability distribution that models the number of failures before a specified number of successes occurs in a sequence of independent and identically distributed Bernoulli trials.
Parameters
The distribution is characterized by two parameters: r (the number of successes) and p (the probability of success in each trial). It can also be expressed in terms of the number of failures k before achieving r successes.
Moment Generating Function
The moment generating function (MGF) of the negative binomial distribution is given by M(t) = [(p e^t)/(1 - (1-p)e^t)]^r. The MGF is useful for finding the moments of the distribution, such as the mean and variance.
Cumulants
Cumulants are related to the moments of a distribution and can be derived from the moment generating function. For the negative binomial distribution, the first cumulant is the mean which can be calculated as r(1-p)/p, and the second cumulant gives the variance.
Applications
The negative binomial distribution is commonly used in various fields such as biology, economics, and quality control to model overdispersed count data.
Geometric distribution, Hypergeometric distribution, Multinomial distribution
Distribution Theory
Geometric Distribution
Geometric distribution models the number of trials needed for the first success in a series of independent Bernoulli trials. The probability mass function is given by P(X=k) = (1-p)^(k-1) * p, where p is the probability of success. It is memoryless, meaning that the probability of success in future trials does not depend on past failures.
Hypergeometric Distribution
Hypergeometric distribution describes the probability of k successes in n draws from a finite population without replacement. It is defined when the total population size N, the number of successes K in the population, and the number of draws n are known. The probability mass function is expressed as P(X=k) = (C(K,k) * C(N-K,n-k)) / C(N,n), where C denotes combinations.
Multinomial Distribution
The multinomial distribution generalizes the binomial distribution for experiments with more than two outcomes. It models the probabilities of obtaining a certain number of each outcome across multiple trials. The probability mass function for a multinomial distribution with parameters n (total trials) and p1, p2,..., pk (probabilities of each outcome) is given by P(X1=k1, X2=k2,..., Xk=kk) = (n! / (k1! k2!... kk!)) * (p1^k1 * p2^k2 *... pk^kk), where k1 + k2 +... + kk = n.
Normal distribution, mean, median, mode, moment generating function, characteristic function, moments
Normal Distribution and Related Concepts
Normal Distribution
Normal distribution is a continuous probability distribution characterized by its symmetric bell-shaped curve. The mean, median, and mode of a normal distribution are all equal and located at the center of the distribution. The spread of the distribution is determined by the standard deviation; a smaller standard deviation results in a steeper curve, while a larger one leads to a flatter curve.
Mean
The mean is the average of a set of values, calculated by summing all values and dividing by the number of values. In the context of normal distribution, the mean represents the center of the distribution and is an important measure of central tendency.
Median
The median is the middle value when a data set is ordered from least to greatest. For normal distributions, the median coincides with the mean, reinforcing the idea that data is symmetrically distributed around the center.
Mode
The mode is the value that appears most frequently in a data set. In a normal distribution, the mode is also equal to the mean and median, as it occurs at the peak of the bell curve.
Moment Generating Function (MGF)
The moment generating function of a random variable is a function that summarizes all the moments (mean, variance, etc.) of the probability distribution of the variable. For a normal distribution, the MGF can be used to derive moments and is defined as E[e^(tX)], where X is a random variable and t is a parameter.
Characteristic Function
The characteristic function is a complex-valued function that provides an alternative way to describe the distribution of a random variable. It is defined as E[e^(itX)], where t is a real number. The characteristic function has properties similar to the moment generating function and can also be used to obtain moments of the distribution.
Moments
Moments are quantitative measures related to the shape of a distribution. The first moment is the mean, the second moment about the mean is the variance, and subsequent moments provide further details about the distribution's shape. In a normal distribution, the even moments (like variance) describe how spread out the data is, while the odd moments (like skewness) indicate symmetry.
Exponential distribution, Gamma distribution, Beta distribution
Distribution Theory
B.Sc. Statistics
Statistics
II
Periyar University
Core Theory IV
Exponential Distribution, Gamma Distribution, Beta Distribution
Exponential Distribution
The exponential distribution is a probability distribution that describes the time between events in a Poisson point process. It is characterized by a constant rate or parameter lambda (λ), which indicates the average number of events in a unit time interval. The probability density function (PDF) is given by f(x;λ) = λe^(-λx) for x ≥ 0. It is memoryless, meaning the probability of an event occurring in the next time unit is independent of how much time has already elapsed.
Gamma Distribution
The gamma distribution generalizes the exponential distribution and is used to model the time until an event occurs. It is characterized by two parameters, shape (k) and scale (θ). The PDF is f(x;k,θ) = (x^(k-1)e^(-x/θ))/(θ^kΓ(k)), where Γ(k) is the gamma function. If k=1, the gamma distribution reduces to the exponential distribution. The gamma distribution can model waiting times for multiple events.
Beta Distribution
The beta distribution is defined on the interval [0, 1] and is parameterized by two shape parameters, α and β. Its PDF is given by f(x;α,β) = (x^(α-1)(1-x)^(β-1))/(B(α,β)), where B(α,β) is the beta function. It is flexible and can take various shapes depending on the values of α and β, making it useful for modeling proportions and probabilities. The beta distribution is widely used in Bayesian statistics.
Functions of Normal random variables leading to t, Chi-square and F-distributions
Functions of Normal Random Variables Leading to t, Chi-square and F-distributions
Introduction to Normal Distribution
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve. It is defined by two parameters: mean and standard deviation. Many statistical methods are based on the assumption of normality.
Transformation to Standard Normal Variable
Any normal random variable can be transformed into a standard normal variable using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. This transformation is essential for deriving other statistical distributions.
Definition of t-distribution
The t-distribution is a family of distributions that are similar in shape to the normal distribution but have heavier tails. It arises when estimating the mean of a normally distributed population in situations where the sample size is small.
Derivation of t-distribution
The t-distribution is derived from the quotient of a standard normal variable and the square root of a chi-square variable divided by its degrees of freedom. It is often used in hypothesis testing and confidence interval estimation.
Chi-square Distribution Explained
The chi-square distribution is a distribution that arises from the sum of the squares of k independent standard normal variables. It is fundamental in various statistical tests, including tests for independence and goodness of fit.
Applications of Chi-square Distribution
Chi-square tests are widely used in categorical data analysis. The distribution helps assess how likely it is that an observed distribution is due to chance.
F-distribution Characteristics
The F-distribution is the ratio of two scaled chi-square variables. It is used primarily in analysis of variance (ANOVA) and regression analysis. The shape of the F-distribution depends on two different degrees of freedom.
Relation Among the Three Distributions
The t-distribution, chi-square distribution, and F-distribution are interconnected. Specifically, the t-distribution can be derived using chi-square distributions and its degrees of freedom, while the F-distribution is based on ratios of chi-square distributions.
Practical Applications in Statistics
These distributions are fundamental in statistical inference, particularly in hypothesis testing, estimation, and regression analysis. Understanding their properties and interrelations is crucial for effective data analysis.
